As with Ashkenazic talmudic interpretation, things really begin with Rashi.
Stage One
Although Rashi considered פשוטו של מקרא ואגדה המישבת דברי המקרא (see his commentary to בראשית ג:ח) to be his major goal/approach על התורה, two of his students, R. Yosef Kara (who had learned in Worms and is the author of many פירושים על הפיוט a well) and Rashbam (Rashi’s grandson and one of the earliest בעלי התוספות) move more toward עומק פשוטו של מקרא. (That is also the case for the fourth member of this group, R. Eliezer of Beaugency whose perushim have survived mainly to נ''ך and is the only member of this group not known as a talmudic scholar in addition.)
The last member of this group is R. Yosef Bekhor Shor, a student of Rabbenu Tam cited in Tosafos, whose פירוש על התורה goes back more to the style of Rashi, integrating פשט and the views of חז''ל. [These five were highlighted in the important early study of Samuel Poznanski (1913), which identified them [and their methods] as the פשטני (צפון) צרפת of the twelfth century.]
Stage Two
Bekhor Shor then “pivots” and becomes the first of the next group moving forward (which was hardly noted by Poznanski--or by anyone else) of בעלי התוספות and other חכמי אשכנז from whom we have some very original פירושים that again seek to integrate פשוטו של מקרא and חז''ל. These include R. Yom Tov of Joigny and R. Ya`akov of Orleans (who like Bekhor Shor were both students of Rabbenu Tam and are cited in Tosafos); R. Yehudah he-Hasid, in the פירוש על התורה that he transmitted to his son R. Zal(t)man; R. Yeshayah di-Trani, in his נימוקי חומש (R. Yeshayah studied with R. Simhah of Speyer and also is the author of תוספות הרי''ד and ספר המכריע, in addition to his תשובות); and R. Moshe of Coucy (בעל הסמ''ג, d. c. 1250).
Versions of the פירוש of ר' יהודה החסיד and נימוקי חומש להרי''ד have been published, but comments of the other figures are found only in manuscript (or in later collections; see below in Stage Three). A sample or two from each, presented in the podcast, provide a window into what they were doing. [And there were also בעלי התוספות like R. Yehi’el of Paris, who do a combination of פשט and דרש.] It is possible that we did not know much until now about the important members of this group and what they produced על התורה because they are following largely in the footsteps of Rashi (and he was the best at all of this). Nonetheless, many interesting and suggestive new פירושים can be found among the members of this group.
Stage Three
The first stage was in northern France during the twelfth century. The second was in both northern France and German (which is something of a חידוש) during the first half of the thirteenth century. The third stage begins around 1240 and produces what are known as Tosafist Torah collections or compilations (קבצי בעלי התוספות על התורה).
The editors or compilers of these collections were often not בעלי התוספות themselves, but they recorded פירושים of the בעלי התוספות on חומש, in quite a few instances where we don’t otherwise have them. The first of these is ספר הג''ן (which is based quite a bit on Bekhor Shor, but directly quotes a number of other בעלי התוספות, both in פשט and in דרש). The best-known among these collections are probably the דעת זקנים and the הדר זקנים (and there is even a third collection known as מושב זקנים which was put together in the fourteenth century). A mention should also be made of פענח רזא and מנחת יהודה (which was compiled in 1313 by an otherwise unknown ריב''א, R. Yehudah b. Eleazar). Lots from Rashi, Rashbam, and Bekhor Shor are quoted in these collections (and even a bit from R. Yosef Qara), but some of these works cite comments from the “stage two” figures as well.
Many of these collections are really a mixture of פשט, דרש (including גמטריה and (רמז and הלכה (or even pieces from Tosafos on Shas). These can sometimes remind us of the גליונות על הפרשה (parshah sheets) that are often given distributed by shuls in Israel, with many of different types of comments in them, so that there is something for everyone within them. Indeed, some aspects of these Tosafos ‘al ha-Torah collections was meant to bring the Torah of the Tosafists (in all these areas) to less learned folks in a simpler or briefer way.
There are also works like R. Avigdor Katz’s פירושים ופסקים על התורה (R. Avigdor was a colleague of Maharam mi-Rothenburg) which provides this kind of mix in a more learned way on the weekly parshah, reminding us a bit of the שאילתות. Note also חזקוני (by R. Hezekiah b. Manoah, c. 1275), which is a real מאסף לכל המחנות of the various peshat approaches, on a high level. Nearly 200 manuscripts of the various בעלי התוספות על התורה collections and compilations are extant, which means that they were quite popular in their day (which is not surprising, given the great variety of materials which they contain), and there are undoubtedly some new insights yet to be discovered with them.
For further reading on all of this (including many primary texts and comments), see Ephraim Kanarfogel, The Intellectual History and Rabbinic Culture of Medieval Ashkenaz (Detroit: Wayne State University Press, 2013), chapters two through four (pp. 110-361); and Ephraim Kanarfogel, “Midrashic Texts and Methods in Tosafist Torah Commentaries,” in Midrash Unbound, ed. M. Fishbane and J. Weinberg (Oxford: Littman Library, 2013), 267-319.
Where do the controversial pirushim like ר יהודה החסיד על התורה fit into this framework?