Combine all ingredients in a bowl. Mix until a soft ball forms. Lightly flour your work surface. Transfer the dough to the floured work surface and continue to knead until soft and elastic, about 5 to 10 minutes. Lightly oil a large bowl. Place the dough in the bowl, cover with plastic wrap, and let rise on your counter-top for 24 hours.
Divide the dough. I cut the dough in half (each weighing 12 1/2 ounces).
Preheat the oven with a pizza stone to 500F. Let the pizza stone heat up for at least 1 hour.
Rolling the dough: Lightly dust your work surface with flour (I like to do this right on the pizza peel so it’s easier to transfer to the oven). Roll the dough into an oval. Mound the cheese on top of the dough, leaving a half-inch border around the edges (used roughly 2 to 2 1/2 cups for each khachapuri). Starting with one side, fold the dough over lengthwise overlapping the cheese a bit (refer to photos above). Repeat with the other side, folding the dough over lengthwise. Pinch the ends to seal, forming a boat shape.
Slide the khachapuri onto the hot pizza stone and bake until the dough is golden brown and the cheese is hot and bubbly, about 12 to 13 minutes. Remove from the oven. With a spoon or fork, make a well in the center of the cheese to make room for an egg. Crack an egg in the well. Brush the dough with egg yolk. Place back into the oven and cook until the egg white is just set but yolk is still runny, about 2 to 3 minutes.
Remove from the oven and add a pat or two of butter to the cheese if you like. With your fork, swirl the molten cheese, egg, and butter together. Cut off a piece of bread and dunk it into the delicious mixture. Enjoy.
