Heat the oven to 425°F/220°C.
In the bowl of a stand mixer fitted with the dough hook, combine the flour, sugar, water, olive oil, and salt. Mix on low speed until the dough comes together. Add the vinegar and knead on medium speed until the dough is smooth, supple, and elastic, 8 to 10 minutes.
Transfer the dough to a lightly oiled bowl, cover, and let rest at room temperature for 1 hour.
Divide the dough into 16 equal pieces, about 3½ to 4 ounces/100 to 110 grams each. Arrange the dough balls in a deep dish, turning each to coat completely in canola or olive oil. Cover and refrigerate for at least 6 hours or overnight.
Meanwhile, make the filling. In a large bowl, combine the feta, mozzarella, egg, spinach, and black pepper until evenly mixed. Taste before adding any salt, as the feta is typically salty enough.
Working with one dough ball at a time, place it on a lightly oiled work surface and gently stretch it with your hands until the dough is paper-thin and nearly transparent.
Spoon about ⅔ cup of the filling along one edge of the dough. Roll tightly to enclose the filling, then shape into a spiral or leave as a long roll.
Arrange the boreks on a parchment-lined baking sheet, leaving a little space between them. Brush with the beaten egg and sprinkle with the sesame seeds.
Bake until deeply golden and crisp, 25 to 35 minutes, rotating the pan halfway through if needed. The bottoms should be evenly browned and crisp.
Let cool for 5 minutes before serving warm.