Probably Turkey. However there is some interesting literature out there on the subject. Some of it argues that burekas have a sort of pre-history as small portable pies eaten by nomads wandering the Asian steppes some 1500 years ago. That theory may or may not be accurate. Burekas are hand pies, and some form of hand pie can be traced to most every wheat eating culture on or near the Eurasian Continent.
Did Culture X invent the hand pie? Or did they just come up with their own word for it? Perhaps they invented the pie and the word. Or were conquered by a people who did. Or perhaps Culture X and Culture Z invented their pies and words independently. One can go mad trying to pin down definitive answers to these sorts of questions. Of all forms of history, food history is by far the most nebulous and speculative. Mostly because the archaeological evidence was long ago digested.
Me, I’m just an armchair food historian, so I’m free to follow my own lines of inquiry in regard these sorts of questions. What interests me about bourekas is the way in which similar words are used to describe similar preparations (i.e. flaky little pies) all around the Mediterranean and Adriatic. You’ve got the burek in the Balkans, the burriche in Italy, the bourek in Greece, the boreg in Armenia, the bureka in Israel, the brik in Tunisia, the boureka in Algeria, the briouat in Morocco, and the bourekita in Spain.
Now me, I take a look at that and think: what do most of those places have in common? If you said the Ottomans, an extra börek for you. The Ottomans had conquered pretty much that entire area (with the notable exceptions of Italy and Spain) by about 1500 A.D.. It seems reasonable to assume that when the Ottomans came, they brought their hand pies with them.
Of course the question can be posed: who introduced the börek to the Ottomans? Possibly no one. The Ottomans’ love of flaky things is well documented. They may well have invented burekas themselves. Or perhaps burekas entered the Empire by way of those West Asian nomads. Or, just perhaps, burekas were introduced to the Ottoman Empire from the other direction, from Spain, by way the Sephardic Jews, who were expelled from Spain in the wake of Ferdinand and Isabella’s infamous Alhambra Decree. They were known to make pies, there were a lot of them (up to 100,000 by some estimates), and they dispersed themselves widely around the Ottoman Empire where they exerted a considerable cultural influence.
Could the Sephardim have brought small, flaky hand pies to the Ottomans? It has been so argued. How much evidence there is for that claim I couldn’t tell you. Based on the flaky dough, I still tend to think that burekas originated in the Middle East or Turkey somewhere. But then I could be wrong, could I not?