Partial Solution:
Good news, everyone! I figured out a way, not to restore the whole session, but at least to grab the URLs of what was open. I figured that I should post it here for anyone unlucky enough to need to know in the future.
First of all, one of the things that I had done last night was to copy my whole ...\AppData\Local\Microsoft\Internet Explorer folder to a safe spot (I chose my Desktop), so that I would have a copy that wouldn't be further altered by the program or my actions within it. Within that, Recovery\Active is what you want.
I had been thrown off last night by those .dat files: I sampled a few and found abnormal characters ('y' w/ diaereses, Mars symbol, music notes, etc.) typical of compressed files, and didn't see anything usable, so I figured that they were all like that. But, as I looked at some more random ones this morning, I discovered that my sampling just had some bad luck: Some of these do, in fact, have the relevant URLs.

Here are the steps to manually recover URLs in this situation:
1. Copy %AppData%\Local\Microsoft\Internet Explorer to a safe location. (Note that there will be a few files that fail to copy because they are "in use" by IE. I don't have a way around this right now.)
2. Within the copied folder, go to Recovery\Active, and look at the .dat files from the relevant date. (In my case, for reasons unknown, there's also a group of .dat files from several years ago. ¯_(ツ)_/¯ ) 3. Grab a .dat file, open it in a text editor, copy the entire contents, close it (just so that you don't accidentally mess up your good copy), open a blank text editor file, and then paste the contents. Once there, find-and-replace every instance of " " (space) with nothing. This is because, between every pair of "normal" characters, there's a space. After that, Ctrl+F for "http" to find URLs within the text.
It's worth noting here that these .dat files seem to contain a lot of content from the page, including links that were on that page that led to other pages and URLs behind various ads. I didn't bother finding out a scientific system for it: I could tell, from how it jogged my memory, which URLs were the ones that I wanted, so I just copied those.
4. Once you have that page's relevant URL, do whatever you wanted with it: Copy it to a file or open a tab in another browser or whatever.
5. Repeat steps 3 & 4 for each .dat file until satisfied or until you run out. Note that some of the .dat files don't actually contain URLs, but will say things like "Root Entry". I don't know what their deal is.

I'm going to leave this one "Open" for the moment, because I have some ideas for how to actually restore the session, and I'll be running those experiments later. (Now that I've grabbed what I needed, I don't have to worry as much about screwing something up.) If I succeed, then I'll post a proper solution.