Make the Dough:
In a large bowl, combine the flour, sugar, yeast, and salt. Add the warm water and olive oil, stirring to form a dough. Knead the dough for 5-7 minutes until smooth and elastic. Cover the dough with a clean towel and let it rise in a warm place for 1 hour, or until it doubles in size.
Prepare the Cheese Filling:
In a medium bowl, mix together the ricotta, mozzarella, and feta cheeses. Set aside.
Shape the Dough:
After the dough has risen, punch it down and divide it into 2 equal portions. Roll each portion into a circle about 1/2 inch thick on a floured surface.
Fill the Dough:
Spread an even layer of the cheese mixture in the center of each dough circle, leaving a small border around the edges. Fold the edges of the dough over the cheese to form a boat shape, pinching the ends to seal the filling inside, leaving a small opening in the center.
Bake the Khachapuri:
Preheat your oven to 475°F (245°C). Place the filled dough onto a parchment-lined baking sheet and bake for 15-20 minutes, or until the crust is golden brown and the cheese is melted and bubbling.
Add the Egg:
Remove the khachapuri from the oven and crack an egg into the center of each one. Return to the oven for 3-4 minutes, or until the egg white is set but the yolk remains runny.
Serve:
Once baked, remove the khachapuri from the oven and brush the crust with melted butter. Serve immediately while hot and gooey. Enjoy!