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

Preheat oven to 220°C (425°F) with a baking stone or tray inside. In a separate bowl, mix the grated Sulguni and Imeruli cheeses.

Punch down the risen dough and divide it into two equal pieces. On a lightly floured surface, roll each piece into an oval shape, about 10x14 inches. Fold and pinch the long edges of each oval to create a 'boat' shape, leaving a cavity in the center.

Transfer the dough boats to a parchment-lined baking sheet. Fill the cavities generously with the cheese mixture. Brush the dough edges with milk (optional, for golden crust).

Bake for 15-20 minutes, or until the crust is golden brown and the cheese is bubbly. Remove from oven, make a small well in the center of the cheese, and crack one egg into each well.

Return to the oven for another 3-5 minutes, or until the egg whites are just set but the yolk is still runny. Remove from oven and immediately place a cube of butter on top of the egg and cheese in each khachapuri.

Serve hot, instructing diners to mix the egg yolk, butter, and cheese together with a fork before tearing off crust pieces to dip.