For the dough

Cup of Yum
1. Grease a large bowl with olive oil and set aside.
2. In the bowl of a stand mixer fitted with a dough hook, combine flour, oil, salt, yeast, sugar and 2⁄3 cup tepid water. Mix on low speed until the dry ingredients come together, about 2-3 minutes. Next increase speed and mix until a smooth, wet dough comes together, about 3-4 minutes.
3. Transfer the dough to the prepared bowl and cover with plastic wrap. Set in a warm place until the dough is doubled in size, 1 hour (See Note 3).

For the filling

1. Combine the cheeses, mixing well, in a large bowl and set aside.

Assembly

1. Place a pizza stone in the center of the oven and preheat to 500°F.
2. Pour the dough onto a lightly floured work surface. Divide into two round pieces. Cover the pieces loosely with plastic wrap and set aside for 15 minutes.
3. On a lightly floured pizza peel or overturned baking sheet, roll one dough piece into a 10 inch circle about 1⁄8" thick. Spread a quarter of the cheese mixture in center of the dough, leaving a 1⁄2" border all around. On one side of the circle, gently roll the dough about a third of the way toward the center. Repeat on the opposite side, leaving a 3" wide center space between the two rolls. Pinch the narrow ends of the rolls together and twist to seal. This creates the Khachapuri's boat shape. Top cheese center with another quarter of the cheese mixture, pressing down lightly.
4. Repeat with remaining dough and cheese. Set the khachapuri aside, uncovered, for 15 minutes, until both are slightly puffed.
5. Beat 1 egg with a teaspoon of water, then brush the dough with egg wash, or just brush with olive oil.
6. Slide the breads onto the hot pizza stone, spacing them at least 3 inches apart. Bake until lightly golden brown and the cheese is melted, 14-16 minutes. Remove from oven and gently press to make an indent in center of cheese (See Note 4) and crack 1 egg into the center of each boat, keeping yolks intact. Return to the oven and continue cooking until the egg whites are just set, 3–4 minutes.
7. Remove the breads, top with a sprinkle of parsley and serve hot. Break bread apart and dunk into the egg and melted cheese.