In a small bowl, dissolve the yeast and sugar in warm water. Let it sit for about 5-10 minutes until frothy.
In a large bowl, mix the flour and salt. Make a well in the center and pour in the yeast mixture and vegetable oil.
Mix until the dough comes together. Turn out onto a floured surface and knead for about 5-7 minutes until smooth and elastic.
Place the dough in a lightly oiled bowl, cover with a damp cloth, and let it rise in a warm place for about 1 hour, or until doubled in size.
In a bowl, combine the shredded mozzarella, crumbled feta, ricotta cheese, egg, black pepper, salt, and chopped parsley (if using). Mix well.
Preheat your oven to 450째F (230째C) and place a baking sheet or pizza stone in the oven to heat.
Punch down the dough and divide it into 2-4 equal pieces, depending on the size you prefer.
On a floured surface, roll each piece into a round or oval shape.
Place the dough on a piece of parchment paper. Spread the cheese mixture evenly over the dough, leaving a border around the edges.
Fold the edges of the dough over the filling to create a boat shape, pinching the ends together to seal.
Brush the edges of the dough with the beaten egg.
Transfer the parchment paper with the Khachapuri to the preheated baking sheet or pizza stone.
Bake for 15-20 minutes, or until the crust is golden brown and the cheese is bubbly and slightly browned.
Remove the Khachapuri from the oven and immediately crack an egg into the center of each Khachapuri. Return to the oven and bake for an additional 3-5 minutes, or until the egg white is set but the yolk is still slightly runny.
Remove from the oven and place a small pat of butter on top of the hot Khachapuri.
Slice and serve hot.
Enjoy your meal!