Doerfer judges it as not originally Turkic, owing to its limited distribution and absence in older texts, and suggests that it more likely came from Iranian languages. So far so good. But, he adds, "Here it also seems to be unetymologizable." Womp womp…
It may be that Andreas (sic) Tietze found something else—I haven't looked at his Turkish etymological dictionary, which was unfinished at the time of his death. If there were a clear answer, however, I don't think it would have escaped Doerfer and the others who looked into this.
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.
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:
I took his class when I was an undergrad. He's a very smart and colorful person who speaks 12 languages. I remember looking forward to each class.
However I think you're missing one of the points of teaching history in schools, which is to instill some sort of shared sense of national identity (which is mostly made up).
I'm Italian, and we are taught that there is some kind of common thread between Caesar, Dante, Machiavelli, and modern Italy. What if we were taught that the durum wheat we use for pasta came from North Africa, that the same idea of making pasta is probably Chinese (by way of the Mongols)? Most of what we consider staples nowadays are either very recent inventions or come from immigration and interaction with our neighbors. How could we justify closing our borders to the refugees coming from the other side of the Mediterranean if we didn't believe the nationalist propaganda?
Another great example of the political use of cuisine is the adaptation of Greek moussaka from its Ottoman ancestor, which involved adding Béchamel to make it more French...
All cultures mimic, adopt and modify aspects of other cultures. See: New York pizza, Australian beer etc. It doesn't follow that the resulting outcome is any less a valid expression of national cultural identity.
Likewise, I don't follow the connection between having a national identity and having national borders.
I once invited my neighbour over for dinner. He introduced me to his BBQ recipe which I now make all the time for my family. But he also got drunk and pissed himself on my couch. I won't be inviting him to move in with us any time soon, despite us having some culinary cultural overlap.
In Europe much of the teaching of history in school has something to do with inventing a common national narrative, that's why everyone minimizes their own war crimes (with some exceptions after Nazism and Fascism, where Germany and Italy wanted a clean cut), and that's why France has forbidden Paths of Glory until 1975 and celebrates WWII victory.
As I was saying in another post almost every country in the Balkans has some kind of national myth about when they were an empire spanning the whole region (so does Italy about the Roman Empire). Ukrainians are very proud about Kievan Rus' (as a tool to prove superiority or precedence over Russians) and I could go on about this.
Maybe New World countries are based less on a nineteenth century national myth, with everyone being able to trace their ancestry to some immigration a few generations back, but here in the Old World this kind of half lies are still very much present.
Re: the connection between national identity and national borders is basically a softer version of the racial substitution conspiracy theory: if these immigrants swamp our country they are going to prevent us from using real pork in carbonara and add pineapple to our pizzas.
In Bosnia, "börek" (or "burek", as is its local spelling) is only filled with meat, while the rest of the dishes fall under the broader category of "pitas" (or pies): sirnica (filled with cheese), krompiruša (filled with potatoes), zeljanica (filled with cabbage), jabukovača (filled with apples)...
In Serbia and Croatia, they're all "burek" with different stuffings, so burek with cheese, burek with potatoes, etc.
It's a weird hill to die on, but you will receive some weird looks if you request just "burek" in Serbia/Croatia/Montenegro, or "burek with cheese" within Bosnia.
Most typically 'me mish' or 'me spinaq' - with meat or with spinach.