Found from the Balkans down through Turkey and the Middle East, the börek is a layered, cheese-stuffed pastry made with phyllo-like yufka. Make Boston chef Ana Sortun's version as a satisfying portable snack, light lunch, or dinner side.
Preheat the oven to 425 degrees F. Butter a 9-inch square baking dish with 1 teaspoon of the melted butter. Lay a yufka sheet into the baking dish, tucking it into the corners with the edges hanging over the sides. (Don't worry if it's not perfect—yufka is very forgiving.) Whisk eggs and egg yolks in a mixing bowl; then whisk in milk, yogurt, salt, and remaining melted butter. Pour ¼ cup of this custard over the prepared yufka. Repeat with a second layer of yufka and custard. Note: Yufka will dry out if it's not used right away. Cover with a clean towel if you're going to finish the dish later.
Break up the mozzarella into bite-size pieces. Reserve ½ cup custard. Distribute ⅓ of the mozzarella over the bottom of the yufka. Fold edges of one of the yufka sheets into the middle, creating a new layer. Add half of the custard and mozzarella, then lay on another sheet of yufka dough. Add the remaining custard and mozzarella, then fold over the edges of the current yufka, followed by the edges of the very first one. Press down with your hand to soak the yufka. Whisk flour into reserved custard and pour over the börek. Using a small knife, score the pastry into squares, like a tic-tac-toe board. Sprinkle with nigella seeds and let rest, 15–20 minutes.
Place börek in the oven and increase heat to 350 degrees F. Bake until börek is set, golden brown, and puffed on top, 50 minutes. Cool 15 minutes.
When cool, cut into 3" x 3" squares. Place a box grater over a bowl and grate the cut sides of a tomato on the largest holes. Discard the skin. Add pepper, olive oil, and 1 teaspoon salt. Stir and set aside. Heat a skillet over low heat. Place the börek slices, cheese side down, into the heated skillet and lightly brown the cheese, 1 minute. Flip and brown the other side, 1 more minute. Serve with tomato purée on the side.