Prepare the Dough: In a large bowl, combine warm water, yeast, and sugar. Let sit for 5 minutes until foamy. Add flour, salt, and olive oil. Mix until a shaggy dough forms.

Knead and Rise: Transfer dough to a lightly floured surface and knead for 8-10 minutes until smooth and elastic. Place in an oiled bowl, cover, and let rise in a warm place for 1-1.5 hours, or until doubled in size.

Prepare Filling: While the dough rises, combine grated Sulguni cheese with softened butter. Mix well.

Shape the Khachapuri: Preheat oven to 400°F (200°C). Divide the risen dough into 4 equal pieces. On a lightly floured surface, roll each piece into an oval or rectangle about 8-10 inches long.

Form the Boats: Along the longer edges of each oval, roll the dough inwards to create a thick crust, pinching the ends to form a boat shape. The center should remain flat.

Fill and Bake (Initial): Place the dough boats on a baking sheet lined with parchment paper. Divide the cheese filling evenly among the centers of the boats. Bake for 15 minutes, or until the crust is golden and the cheese is melted and bubbly.

Add Eggs and Finish Baking: Remove from oven. Gently crack one egg into the center of the cheese in each boat. Return to the oven and bake for another 3-5 minutes, or until the egg whites are set but the yolk is still runny.

Serve: Immediately upon removing from the oven, place a pat of butter on top of each egg yolk. Serve hot, stirring the butter and yolk into the cheese before eating.