For the Dough:

1. Preheat the oven to 425°F. Set out three baking sheet pans. Place a parchment sheet or silicone mat on each pan. Set aside.
2. With your hands or a large spoon, mix and mash the cheeses together in a medium bowl. Add a pinch of salt and pepper and mix until combined.
3. Very lightly flour your work surface. Turn the dough onto the floured surface and cut it into equal thirds. Mold the divided pieces gently back into rough rounds. Cover the rounds with a slightly damp towel. Let rest for 10 minutes.
4. Working with one round at a time, roll out the round with a rolling pin to make a rough oval shape, about 12 inches long and 7 inches wide. Transfer to one prepared sheet pan.
5. Roll both of the long sides tightly towards the center of the dough until there is about 1 inch separation between the rolled edges. Pinch the top and bottom ends together and twist to seal tightly. Gently manipulate and stretch the rolled sides to make a shape like a bulbous canoe, while ensuring the seal ends are still tight. Let rest for 10 minutes.

For Baking and Serving: 

1. Spoon one third of the cheese mixture in the center of the shaped dough, spreading to the crust. Brush the sides with just a little olive oil, and sprinkle the top with a pinch of salt and pepper.
2. Repeat with the other two pieces of dough, and with the remaining cheese.
3. Transfer the pans to the preheated oven. Bake until the dough crust is lightly golden-brown, but still slightly underbaked, about 10 minutes, rotating the pans halfway through baking. Remove from the oven.
4. With a spoon, create a deep well in the center of the cheese large enough for one egg by moving the cheese towards the sides. Pour one cracked egg in the middle of each khachapuri and return to the oven. Bake until the white is just set around the edges, about 5-6 minutes.
5. Remove from the oven, place 1 tablespoon of butter on each khachapuri, and sprinkle with salt and pepper.
6. Serve immediately – stir the egg and butter into the hot cheese. Tear off the outside crust to dip into the cheese and egg mixture. Enjoy!