Whisk milk, sugar, and yeast in a bowl. Let it stand for 10 minutes until foamy.
Add the flour, salt, and add oil. Stir to combine.
Knead the dough for 5 minutes until very soft and elastic.
Divide dough into two balls, brush with oil, cover, and let proof for 1.5 hours.
Preheat oven to 400 F.
Mix mozzarella and queso fresco in a bowl, add a splash of cream, and season with salt.
Roll each dough ball into a 10-inch oval. Place dough on a lined baking sheet. Spread the cheese mixture over the dough, leaving a border. Fold and pinch edges to seal, forming a boat shape.
Flip the khachapuri over and cut a center slit—roll edges to form a boat shape and fill with remaining cheese. Let rest for 15 minutes.
Bake for 18-20 minutes until golden brown. Make a well in the center, add an egg yolk, and top with butter. Serve
