How to Make It
Heat milk in a saucepan over medium to 115°F; pour into a bowl. Sprinkle with yeast and 1 teaspoon of the sugar. Let stand 1 minute; stir. Let stand until foamy, about 10 minutes.
Combine flour, softened butter, 1 teaspoon of the salt, and remaining 2 tablespoons sugar in the bowl of a heavy-duty electric stand mixer fitted with the dough hook attachment. Beat on low speed until butter is incorporated. With mixer running, add yeast mixture. Increase speed to medium, and beat until a supple dough forms, about 10 minutes.
Turn dough out onto a work surface, and shape into a ball. Place oil in a large bowl.Place dough ball in bowl, and turn to coat completely with oil. Cover bowl loosely with a clean towel, and place in a dry, warm place. Let stand until dough has doubled in size, about 1 hour.
Turn dough out onto a work surface, and divide into 6 even portions. Roll each portion into a ball. Place balls on a baking sheet, and let stand in a dry, warm place until slightly springy, about 20 minutes.
Preheat oven to 425°F. Stir together mozzarella, feta, and cottage cheese in a large bowl.
Lightly dust a work surface with flour. Working 1 dough ball at a time, roll into an oval shape with a maximum length of 10 inches and maximum width of 4 inches. Spoon 1/2 cup cheese mixture into center of oval, leaving a 1-inch border. Roll dough borders up and toward center of oval, and pinch dough together at ends to create a canoe shape.Repeat with remaining dough balls and cheese mixture.
Line 2 baking sheets with parchment paper, and coat with cooking spray. Place 3 khachapuri on each prepared baking sheet. Brush crusts evenly with 1/3 of the melted butter. Bake in preheated oven 15 minutes.
Remove baking sheets from oven. Break an egg in the center of each khachapuri. Brush crusts evenly with 1/3 of the melted butter. Return to oven, and bake until dough is golden brown and egg whites are set. Remove from oven. Sprinkle with pepper and remaining 1 teaspoon salt, and brush evenly with remaining butter. Serve immediately.