In a small bowl, combine the yeast with warm water and sugar. Set aside for about 10 minutes until it becomes bubbly and foamy.

In a large mixing bowl, combine the flour and salt. Add the milk, egg, and the activated yeast mixture, mixing thoroughly. Begin kneading the dough, either by hand or in a stand mixer, and gradually add the oil while kneading. Continue until the dough is smooth and elastic, about 8-10 minutes by hand or 5-6 minutes in a mixer. Place the dough in a greased bowl, cover with a damp cloth, and let it rise in a warm place for 1 hour or until doubled in size.

In a medium bowl, mix the shredded mozzarella and crumbled feta. Add the egg and mix until thoroughly combined.

Preheat the oven to 220°C (430°F). Punch down the risen dough and roll it into an oval, about ⅓ inch (1 cm) thick. Optional: Place a thin layer of cheese filling along the edges of the dough before shaping. Place the cheese filling in the center, leaving space around the edges. Roll up the sides toward the center to create a boat shape, pinching the ends to secure.

Transfer the khachapuri to a baking sheet lined with parchment paper. Brush the edges of the dough with egg wash. Bake in the preheated oven for 15-20 minutes until the dough is golden and cheese is bubbly. For an optional egg topping, crack an egg into the center of the bread about 5 minutes before the end of baking and bake until the egg white sets.

Remove from the oven and brush the exposed crust with melted butter. Serve warm.