Sevan Nişanyan (an Armenian-Turkish language researcher) suggests that it may be of Persian origin (būrak, a dish made with dough and meat). He, however, doesn't dismiss the possibility that it may have some etymological relationship with the Slavic word peirogi (pirog пирог, pirojki пирожки - a dough dish) and the Persian word having a Turkic root (having made a full circle like tülbent).
I love Nişanyan Sözlük! It's an incredible resource.
Anyway, Doerfer mentions the pierogi theory, attributing it to Ramstedt (see the last part of the screenshot I posted), but he says it's been refuted by Vasmer. (Presumably Max Vasmer? Wow, that guy had a hell of a career.)
There is the impression of something circular going on. If you look for būrak in the Dehkhoda Persian dictionary, you'll see that it was also known as a kind of yogurt soup/stew, with the name perhaps originating as an alternate form of the Turkic (!) bughrā. That, in turn, may be an abbreviation of bughrā-khānī, which you can find described, e.g. in Steingass's dictionary, as a pastry dish dressed with milk or gravy. (It was allegedly named after a ruler of Khwarazm.)
So, if you wanted to piece the whole thing together, you could theorize that bughrā-khānī was shortened to bughrā, from which the Persian būrak eventually derived—and then went back into Turkic as börek. This could also involve flexibility in the intended dish, between a stuffed pastry with gravy, and some kind of soup… maybe with dumplings? It wouldn't be the strangest instance of metonymy.
You know, it's crazy for me to revisit this topic. Some time ago I was trying to figure out a sixteenth-century Persian poem, in which there are references to bughrā'ī and māhīchah. The best I could figure out, it was supposed to mean a soup with pieces of pastry in it. It never occurred to me that this might be connected to a theorized etymology of börek.
Etymology can be addictive. The links, theories and how the words reflect the evolution of societies always amaze me.
Nişanyan also has a blog where he goes deep in words he finds interesting. It's in Turkish and translators fail horribly, unfortunately, to keep most of the meaning. I still wanted to share if anyone speaks the language:
https://www.nisanyansozluk.com/?k=b%C3%B6rek&view=annotated