In a large bowl, combine flour, yeast, sugar, and salt. Gradually add warm water or milk and olive oil, mixing until a soft dough forms. Knead on a lightly floured surface for 5-7 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.

Divide the dough into 4 equal portions. On a lightly floured surface, roll each portion into an oval shape (approx. 8x12 inches). Fold and pinch the two long edges of each oval to form a boat-like crust, leaving a cavity in the center for the cheese.

Preheat oven to 220°C (425°F). In a separate bowl, combine the grated Sulguni and crumbled Imeruli cheeses. Fill the dough boats generously with the cheese mixture.

Place the filled Khachapuri boats on a baking sheet lined with parchment paper. Bake for 15-20 minutes, or until the crust is golden brown and the cheese is bubbly. Remove from oven.

Carefully crack one egg into the center of the cheese filling of each Khachapuri. Return to the oven and bake for another 3-5 minutes, or until the egg whites are set but the yolk is still runny. Remove from oven.

Immediately place a pat of unsalted butter on top of the egg in each Khachapuri. Serve hot, stirring the egg and butter into the molten cheese before diving in.