I started Moshe Weinfeld's Deuteronomy and the Deuteronomic School. Here are three items:
1. According to Weinfeld, there are Deuteronomistic speeches in the Hebrew Bible (and it's not surprising that the Deuteronomist has a propensity for speeches, considering that Deuteronomy is one long speech by Moses to Israel). For Weinfeld, the purpose of these speeches was "to demonstrate that the fall of Judah and the destruction of the temple, the most overwhelming of catastrophes in Israelite history, occurred, like the fall of the kingdom of Israel, by force of the prophetic word of God" (page 28). Or the Deuteronomistic speeches had other agendas, such as highlighting God's punishment of the sin of idolatry. For example, the Deuteronomist takes I Kings 21---which is about Jezebel's murder of Naboth, and which affirms that Ahab's blood will be "shed in the very place where he had shed Naboth's blood"---and puts into Elijah's mouth a prophecy about the destruction of Ahab's line for his cultic pollutions and his "propagation of the sins of Jeroboam" (page 18). The Deuteronomist makes the Naboth story about more than God's punishment of Ahab for murdering an individual.
Weinfeld also thinks that some of Jeremiah's prose sermons are from the hand of the Deuteronomist---which would mean that some of Jeremiah's condemnations of Judah and prediction of her destruction were retrospectively put into the mouth of Jeremiah by the Deuteronomist. Weinfeld believes that Jeremiah 17:19ff. was inserted by the Deuteronomist, for "Not only does it contradict the original prophecy (vv. 1 and 10) according to which Jeremiah was to deliver his message only before the group of senior priests and the elders of the people, but it also contains the illogical suggestion that the prophecy in reality could address several 'kings of Judah' at one and the same time" (page 29). I'm not sure if Weinfeld is saying that the Deuteronomist just has a knack for being illogical, and that's why we know that Jeremiah 17:19ff. is from him, but his point may be that the Deuteronomist had a "big-picture" view of Judah and her sins---of which previous kings were culpable. Weinfeld contrasts Jeremiah 17:19ff. with Jeremiah 4:9, which he considers to be authentically Jeremian. Jeremiah 4:9 talks about only one king, saying that his heart will perish in Judah's time of peril. Apparently, Weinfeld is not suggesting that all of the predictions about Judah's destruction are from the retrospective hand of the Deuteronomist, for he does hold that some of them go back to Jeremiah. But his view is that some of the speeches attributed to Jeremiah are actually from the Deuteronomist.
Weinfeld also sees the Deuteronomist's speeches as historically implausible---in terms of describing the conditions of Jeremiah's day. On pages 29-30, Weinfeld states:
"The sins which are mentioned in [Jeremiah 17:19ff.] (the burning of incense to foreign gods, the shedding of innocent blood, the building of high places to Baal, the burnt offering of child victims) recur in stereotyped formulations in other deuteronomic sermons in Jeremiah. They are the sins of which the deuteronomic editor accuses Israel and Judah throughout their history from the time they left the land of Egypt until the 'present day' (Jer. 7:25; 11:7; 32:30-1, see also I Sam. 8:8; 2 Kgs. 21:15), the sins which Yahweh's servants perennially inveighed against but to no avail (7:13, 15; 11:8; 25:3, 4, etc.). It is highly improbable that these sins ever flourished in Judah after the Josianic reform. The Tophet, for example, was destroyed by Josiah (2 Kgs. 23:10) and is most unlikely to have been reintroduced after his death. It is likewise difficult to suppose that Baal worship, which was characteristic of the northern kingdom (see Jer. 23:13), ever took root again in Judah after the radical extirpation of foreign worship in the days of Josiah. It would appear therefore, that most of these sins, and particularly the sins of the Tophet and the burning of incense to foreign gods, are those past offenses which the Deuteronomist accused Manasseh of encouraging and which he regarded as having inevitably sealed Judah's doom (2 Kgs. 21:1-16, see also Jer. 15:4). Had Jehoiakim and his successors also been guilty of such sins there would have been no need for the Deuteronomist to blame Manasseh in particular for the fall and destruction of Judah."
Jeremiah 17:19ff. doesn't describe these sins, as far as I can see, but it does discuss the Sabbath. Jeremiah 7, however, does refer to them, and Weinfeld regards the sermon there as Deuteronomistic.
Weinfeld compares the Deuteronomist's insertion of speeches into the Hebrew Bible with Greek historians like Herodotus and Thucydides. Thucydides admitted that "it was his habit to make the speakers say what in his opinion was demanded of them by the various occasions (I.22.I)" (page 52).
2. Unlike Gerhard Von Rad and others, Weinfeld does not think that Levites wrote Deuteronomy. Von Rad thought that the Levites were the authors because Nehemiah 8 and II Chronicles 17:7-9 presents them as teachers, whereas Weinfeld maintains that such was the Chronicler's idea, not something that can be projected back onto pre-exilic times. Moreover, for Weinfeld, if Levites wrote Deuteronomy, they would be "cutting off the branch on which" they sit, for Deuteronomy advocates a centralization that deprived Levites of their positions at sanctuaries as well as regards the Levites as a "personae miserabiles" (page 55). (But I'm curious as to how Weinfeld will address Deuteronomy 18's inclusion of the Levites from outside the gate into the worship at the central sanctuary.) Weinfeld also doubts that "so insignificant a provincial class as the Levites" could have had access to the "rich variety of material" that was used to create Deuteronomy and the Deuteronomistic History: cultic works and "a vast body of literary material, including historical memoirs, popular narratives, historical annals, chronicles, administrative lists, etc." (page 55).
3. Weinfeld compares the Deuteronomic covenant with ancient Near Eastern vassal treaties, which had such things as the agreement of the vassals to show undivided love to the king (not to other kings), commands, an "invocation of witnesses", and blessings and/or curses (page 61). Copies of the agreement were made for each party, and they were placed in the sanctuary (as the tablets containing the Decalogue were put inside of the Ark of the Covenant). The debate among scholars concerns which vassal treaties the Israelite covenant reflects. Many maximalists argue that the Israelite covenant resembles second millennium B.C.E. Hittite vassal treaties---and, while they have arguments for that (as we shall see), an underlying motivation for their preference of a second millennium date is that, according to the Hebrew Bible, the revelation at Sinai or Horeb was made in the second millennium B.C.E., and so the historicity of the Hebrew Bible's narrative on the covenant's origin is bolstered when the narrative is demonstrated to reflect a second millennium concept. Other scholars, however, have argued that vassal treaties also existed in the first millennium B.C.E., and so the Israelite covenant could have originated long after the time that the Hebrew Bible depicts. Weinfeld tends to go with the latter argument, as he speculates that the Deuteronomic covenant originated as a protest against the Assyrians: the Assyrians regarded Judah as a vassal who was to serve them alone, but Josiah was declaring that Judah's loyalty was only to YHWH---appealing to a document (Deuteronomy) that resembled a vassal treaty. At the same time, Weinfeld compares Deuteronomy with ancient Near Eastern vassal treaties in general---Hittite, Assyrian, etc.---rather than just focusing on the Assyrian vassal treaties.
In Ancient Israel in Sinai, on page 192, maximalist James Hoffmeier contrasts the Assyrian-Aramean vassal treaties with the Hittite ones, contending that the Israelite covenant resembles the latter (from the second millennium) rather than the former (from the first millennium):
"The first-millennium treaties lack a historical prologue, and the deposition/reading provision is absent. Furthermore, the witnesses stand early in this period, whereas they are fifth in the second-millennium formula, and there are no blessings. As a consequence of these omissions and differences in order, it is illogical to believe that the six-point structure of the Sinai and Deuteronomy treaty texts originated from the four-point treaty formula of the first millennium."
Weinfeld actually wrestles with these issues, on pages 67-69. Weinfeld thinks that the Deuteronomic covenant overlaps with the Assyrian vassal treaties rather than the Hittite ones because the Deuteronomic covenant contains elaborate curses, like the Assyrian vassal treaties, whereas the curses of the Hittite treaties are "very short and generalized". But Weinfeld acknowledges that the Hittite treaties have a historical prologue, like the Deuteronomic covenant, whereas the Aramean-Assyrian ones do not. Weinfeld entertains the possibility that pieces of the first millennium treaties are missing---for three of the five first millennium treaties that have been found "have been mutilated at the beginning just where the historical prologue ought to be." Weinfeld also says that "in a recently published fragment of a neo-Assyrian treaty there does seem to have been a remnant of a historical prologue." Moreover, there is a historical prologue in neo-Assyrian land grants, showing that the Assyrians knew about the idea of a historical prologue, whether it's in their treaties or not.
But Weinfeld then shifts gears and portrays the Assyrians as strict: their treaties did not have blessings, and they did not express love for the vassals, whereas the Hittite treaties express the sovereign's love for the vassals as well as "justify their demands for loyalty [and] give promises of help in time of danger, as well as to bestow divine blessings for loyal service." The Assyrians, however, felt those things to be beneath them, and they preferred to rule by inspiring fear in their vassals (thus the long, elaborate curses). There is some information in Weinfeld's work that slightly contradicts this picture, however, for the Assyrians gave land-grants as a reward for loyal service (page 75), and Ashurbanipal criticizes Gyges of Lydia for trusting in his own strength and not obeying the Assyrian god Ashur (page 85), which seems to imply that the vassal Gyges should trust in the protection of Assyria rather than his own power. Even if the idea of blessing was not in the Assyrian treaties, it could have been part of how Assyria related to people and nations, and so I don't think that we can rule out Deuteronomy being based on concepts from the first millennium B.C.E.