Khachapuri (Georgian Cheese Bread)

Dough: Combine water and yeast in a bowl; let stand until foamy, about 5 minutes. Add flour, salt, and sugar; stir until a ragged dough forms. Transfer to a lightly floured surface; knead with floured hands until smooth, elastic, and tacky but no longer sticky, about 5 minutes. Transfer to a medium bowl brushed with oil, turning dough to evenly coat. Cover with plastic wrap and let stand in a warm, draft-free place until doubled in volume, 1 1/2 to 3 hours; or refrigerate until doubled in volume, at least 12 hours and up to 3 days (return to room temperature before using, 1 to 2 hours.)

Filling: Preheat oven to 500 degrees, with center rack lined with a pizza stone or baking sheet. Divide dough in half. Place one half of dough on a lightly floured 8-by-13-inch sheet of parchment. Roll into a roughly 7-by-12-inch oval. Stir together feta and mozzarella. Pack half of cheese mixture into center of oval, leaving a 1-inch border. Fold edges of dough over cheese mixture; pinch ends together and twist to create a boat shape. Repeat with remaining half of dough and cheeses.

In a small bowl, whisk egg white with a fork. Brush over crusts. Slide breads (still on sheets of parchment) onto stone. Bake until cheeses are melted and crusts are well browned, 16 to 20 minutes. Drop 1 egg yolk into center of each bread. Bake until just warmed through, about 1 minute. Using a large flat spatula, remove breads from oven and transfer to a wire rack. Brush crusts with melted butter and divide remaining 2 tablespoons room-temperature butter between fillings; let melt. Sprinkle with flaky salt and pepper. Let cool slightly; serve.