Adjarian Khachapuri

Step 1
Whisk together milk, yeast, and sugar in a large bowl. Let stand, covered with a kitchen towel, until mixture is foamy, about 10 minutes.

Step 2
Sift flour into milk mixture. Add oil and 3/4 teaspoon salt; stir with a rubber spatula until mixture is well combined and a shaggy dough forms. Transfer to a work surface and shape into a ball; knead until smooth and soft but slightly sticky, about 5 minutes.

Step 3
Shape dough into a ball and lightly brush with oil; transfer to a large bowl and cover loosely with plastic wrap. Let rise in a warm (70 to 80 degrees F (21 to 26 degrees C)) place until doubled in size, about 1 1/2 hours.

Step 4
Stir together mozzarella and ricotta in a small bowl. Season with remaining pinch of salt; stir to combine.

Step 5
Divide dough into 4 equal portions. Roll each into a 5×8-inch oval, about 1/4 inch thick, on a lightly floured work surface. Spread about 1/4 cup cheese mixture over each oval, leaving a 11/2-inch border uncovered. Fold the long edges of each oval inward to meet at the center and pinch the seam together tightly to seal in cheese mixture. Turn dough over, so the seam side is down. Use a small, sharp knife to make a shallow lengthwise slit down the center of the top layer of dough, stopping within 1 inch of each end. Use your fingers to roll each long edge away from the slit and tuck it under to create a rolled border, sealing in some of the cheese mixture.

Step 6
Evenly divide and spread remaining cheese mixture (about 1/4 cup each) in the centers of each flatbread, lightly pressing into dough. Arrange flatbreads 2 to 3 inches apart on 2 parchment-lined baking sheets. Cover with a kitchen towel in a warm place (70° to 80°F) until slightly puffed, 25 to 30 minutes.

Step 7
Preheat the oven to 400 degrees F (200 degrees C).

Step 8
Bake, rotating baking sheets halfway through, until edges are golden brown, 20 to 25 minutes. Use a spoon to make a well in the center of the filling in each flatbread. Put an egg yolk in each well. Bake until yolks are thickened but still runny, 30 to 40 seconds more.

Step 9
Transfer flatbreads to a serving platter. Top each with 1/2 tablespoon butter. Stir yolks into melted cheese. Serve hot.