Place a pizza stone in the oven on the bottom rack and heat the oven to 500ºF (260ºC).
Turn the dough out on a lightly floured surface. Divide it in half. Press and stretch each half of the dough into a 13” oval and place on a piece of parchment paper.
Working one at a time, roll the long ends of the dough towards the middle, pinching and pressing the dough to form walled sides. Then pinch the short ends of the dough together on either side to form a boat.
In a small mixing bowl mix together the mozzarella, feta and ricotta cheeses until well combined. This can also be done quickly in a food processor.
Divide the cheese mixture between the two cheese boats.
Cover the khachapuri with plastic wrap and let rise in a warm place for an additional 20 minutes.
In a small bowl whisk one of the eggs with 1 tablespoon of water. Brush the edges of the dough with the egg wash.
Working one at a time, use a pizza peel to move the bread boat to the oven on the pizza stone. Bake until the edges of the dough are golden brown and the cheese has melted, about 15 to 18 minutes minutes.
Crack an egg into a ramekin, discarding the whites. Use a spoon to press a well into the center of the cheese. Carefully add an egg yolk into the center of the khachapuri and add pats of butter. Repeat with the remaining khachapuri.
Serve hot. To eat, use a fork to mix the egg yolk into the center, tear the bread from the sides and dip in the center.