In a large bowl, combine flour, yeast, sugar, and salt. Gradually add lukewarm water and olive oil, mixing until a shaggy dough forms. Transfer to a floured surface and knead for 8-10 minutes until smooth and elastic. Place the dough in a lightly oiled bowl, cover with plastic wrap, and let it rise in a warm place for 1-1.5 hours, or until doubled in size.

Preheat your oven to 450°F (230°C). Lightly flour a baking sheet or line with parchment paper.

While the dough rises, prepare the cheese filling by combining shredded mozzarella, crumbled feta, and ricotta (if using) in a medium bowl.

Once the dough has risen, punch it down gently and divide it into 4 equal portions. On a lightly floured surface, roll each portion into an oval shape, about 1/4 inch thick.

Spread a quarter of the cheese filling evenly over the center of each oval, leaving a border of about 1 inch around the edges. Roll the long edges of the dough inwards towards the center, creating a boat shape. Pinch the ends together firmly to seal and form points.

Carefully transfer the Khachapuri boats to the prepared baking sheet. Whisk one large egg and brush it over the exposed dough edges to create a golden crust.

Bake for 12-15 minutes, or until the dough is golden brown and the cheese is bubbly. Remove from the oven.

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

Remove from the oven, immediately place 1 tablespoon of butter onto the egg yolk of each Khachapuri. Serve hot, allowing diners to mix the egg and butter into the cheese before eating with pieces of the crust.