Step 1

For the dough: Lightly oil a medium bowl. Heat the milk with 1/2 cup water in a small saucepan over low heat to between 110 and 115 degrees F. Transfer it to a small bowl and stir in the sugar, then sprinkle over the yeast and let stand until foamy, 5 to 10 minutes. (Cover the bowl if the room or the yeast is cold.)

Step 2

Put a pizza stone on the lowest rack in your oven, removing the second rack if there is one. Preheat the oven to 500 degrees F. Cut two 10-by-7-inch pieces of parchment paper.

Step 3

Put the flour in a medium bowl and whisk in the salt. Make a well in the center and pour in the yeasty mixture and the oil. Stir with a wooden spoon until the flour is incorporated and the mixture forms a shaggy ball -- it will be soft and sticky. Flour your work surface, scrape the dough onto it and knead until smooth, elastic and still slightly sticky, about 5 minutes. Put the dough in the oiled bowl, turn it to coat with oil and cover the bowl loosely with plastic wrap. Set the bowl in a warm place until doubled in size, 1 hour to 1 hour and 15 minutes.

Step 4

For the filling and topping: Work the mozzarella and feta together with a fork in a medium bowl. Make a little well in the center and add 1 egg. Beat the egg a bit with a fork, and mix and mash well to combine with the cheeses.

Step 5

Lightly flour your work surface. Turn the dough onto the floured surface and cut it in half. Round the halves gently back into rough rounds. Put 1 piece of parchment paper on your work surface and flour it lightly. Press and use a rolling pin to roll 1 piece of dough into a round 1/8-inch thick and 10-inches across. Slide it onto the paper -- it will hang over the long edges -- and then slide it, paper and all, onto the back of a baking sheet. (Alternatively, use a pizza peel.) Repeat with the other piece of dough, transferring it to the back of a second baking sheet.

Step 6

Spoon half of the cheese mixture on the first round, spreading some 1/2-inch from the edge and mounding the rest in the middle (there will be a big mound). Roll the sides up tightly to make a shape like a cheese-filled canoe -- it should be about 10 inches long and 4 inches wide. Pinch the ends together and twist to seal tightly. Repeat with the other piece of dough and remaining cheese. Let rest 10 minutes.

Step 7

Slide the khachapuris on their papers onto the pizza stone, leaving a few inches of space between them. Bake, turning them 180 degrees after about 8 minutes, until golden and crisp, about 12 minutes total. Crack 1 egg in each khachapuri and continue to bake until the white is just set around the edges, but some of the white and the yolk are still a bit raw, about 3 minutes. Remove to a serving platter, dot each khachapuri with 1 tablespoon butter and stir the eggs into the hot cheese¿they will continue to cook. Serve immediately, tearing off pieces of bread to scoop up the hot, buttery, eggy, gooey cheese.