Georgian Cheese and Egg Bread (Acharuli Khachapuri)

Combine yeast, sugar, and 2⁄3 cup water heated to 115° in a bowl; let stand until foamy, about 10 minutes. Add oil, flour, and salt; mix with a wooden spoon until a soft dough forms. Transfer to a lightly floured surface and knead until smooth and elastic, about 4 minutes. Transfer to a lightly greased bowl and cover loosely with plastic wrap; set in a warm place until doubled in size, about 45 minutes.

Place a pizza stone on a rack in lower third of oven. Heat oven to 500° for 1 hour. Combine cheeses in a bowl; set aside. Punch down dough and divide in half. On a piece of lightly floured parchment paper, roll half of dough into a 10" circle about 1⁄8" thick. Spread a quarter of the cheese mixture over dough, leaving a 1⁄2" border. On one side of the circle, tightly roll dough about a third of the way toward the center. Repeat on the opposite end. There should be a 2"-3" space between the two rolls. Pinch the open ends of the rolls together and twist to seal, making a boat shape; place another quarter of the cheese mixture in the middle; repeat with remaining dough and cheese. Transfer boats on paper to stone; bake until golden brown, 14–16 minutes. Crack 1 egg into the center of each boat. Return to oven until egg white is slightly set, 3–4 minutes. Place 2 tbsp. butter on each bread; serve hot.