Prepare the Dough: In a large bowl, combine warm milk, yeast, and sugar. Let sit for 5-10 minutes until foamy. Add flour, salt, and olive oil. Mix until a shaggy dough forms, then knead on a floured surface for 5-7 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 the Filling: In a medium bowl, combine shredded mozzarella, crumbled feta, ricotta, and 1 egg. Mix well.

Shape the Khachapuri: Preheat oven to 450°F (230°C). Divide the risen dough into 4 equal portions. On a lightly floured surface, roll each portion into an oval shape (about 8x12 inches). Spoon a quarter of the cheese filling down the center of each oval.

Form the Boat: Fold the longer sides of the dough over the filling, pinching the edges together to form a 'boat' shape with open ends. Pinch the ends of the boat to seal them, creating a pointed tip.

Bake: Carefully transfer the khachapuri boats to a baking sheet lined with parchment paper. Bake for 15 minutes, or until the crust is golden brown.

Add Topping & Finish: Remove from oven. Make a well in the center of the cheese filling in each boat and crack one egg into each well. Return to the oven and bake for another 3-5 minutes, or until egg whites are just set but the yolk is still runny.

Serve: Remove from oven. Immediately place a pat of butter into each egg yolk. Serve hot, mixing the egg, butter, and cheese together with a fork before tearing off pieces of the crust to dip into the creamy center.