1. In a large bowl, combine warm milk, yeast, and sugar. Let sit for 5-10 minutes until foamy.

2. Add flour, salt, and olive oil to the yeast mixture. Mix until a shaggy dough forms, then 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 60-90 minutes, or until doubled in size.

3. While the dough rises, prepare the cheese filling: In a bowl, combine shredded Sulguni, Imeruli, and cottage cheese. Mix well.

4. Once risen, punch down the dough and divide it into 4 equal pieces. On a lightly floured surface, roll each piece into an oval shape.

5. Gently roll and pinch the long edges of each oval towards the center to form a boat shape, leaving the center open for the cheese. Pinch the ends tightly to seal.

6. Fill the center of each dough boat generously with the cheese mixture.

7. Carefully transfer the khachapuri boats to a baking sheet lined with parchment paper. Brush the exposed dough edges with an egg wash (1 egg yolk whisked with 1 tablespoon water).

8. Preheat oven to 400°F (200°C). Bake for 15-20 minutes, or until the crust is golden brown and the cheese is melted and bubbly.

9. Remove from the oven. Carefully crack one egg into the center of each khachapuri, on top of the melted cheese. Return to the oven for another 3-5 minutes, or until the egg white is set but the yolk is still runny.

10. Remove from oven and immediately place a pat of butter on top of the hot egg and cheese. Serve hot, stirring the egg, butter, and cheese together before tearing off pieces of crust to dip.