In a large bowl, combine flour, sugar, yeast, and salt. Gradually add warm water and olive oil, mixing until a shaggy dough forms. Turn onto a lightly floured surface and knead for 8-10 minutes until smooth and elastic. Place dough in a greased bowl, cover, and let rise in a warm place for 1-1.5 hours, or until doubled in size.

Preheat oven to 450°F (230°C). Prepare a baking sheet with parchment paper.

Make the filling: In a medium bowl, stir together the mozzarella, feta, ricotta, 1 egg, and 1 tablespoon flour until thoroughly combined. Divide into 4 sturdy balls and set aside.

Punch down the risen dough and divide into 4 equal pieces. On a lightly floured surface, roll one piece into an oval shape, about 1/4-inch thick. Take one cheese ball and shape it into a log along the long edges of the oval dough.

Fold the edges of the dough over the cheese logs and pinch firmly to seal, creating a cheese-stuffed crust. Bring the two sealed edges together at the ends and pinch firmly to form a boat shape. The center should be open.

Carefully transfer the khachapuri boat to the prepared baking sheet. Repeat with the remaining dough and filling. Fill each boat with a quarter of the remaining cheese filling.

Bake for 12-15 minutes, or until the crust is golden brown and the cheese is bubbly. Remove from oven. Crack one egg into the center of each khachapuri.

Return to the oven and bake for another 3-5 minutes, or until the egg white is set but the yolk is still runny. Keep a close eye on it to prevent overcooking the yolk.

Remove from oven. Place a tablespoon of butter onto the hot cheese and egg in the center of each khachapuri. Serve immediately, instructing diners to stir the butter and egg yolk into the cheese before eating.