How to Make It
In a large bowl, whisk the yeast with the sugar and 1 cup of warm water. Let stand until foamy, about 10 minutes. In a small bowl, whisk the milk with the egg and 3 tablespoons of oil.
Add the milk mixture, the flour and salt to the foamy yeast and mix until a dough forms. Form the dough into a ball and transfer to a lightly oiled large bowl. Cover the bowl with plastic wrap and let the dough rise in a warm place until doubled in bulk, about 1 1/2 hours.
Line 2 baking sheets with parchment paper. On a lightly floured work surface, divide the dough into 4 equal pieces and form into smooth balls. Set 2 balls of dough 5 inches apart on each sheet. Cover loosely with plastic wrap and let rest in a warm place for 45 minutes.
In a blender, puree the feta with 2 of the eggs, 2 tablespoons of water and the salt. Transfer the mixture to a bowl and stir in the mozzarella.
Preheat the oven to 450°. Transfer the dough balls to a lightly floured work surface. Line the baking sheets with clean parchment. Roll out 1 ball of dough to a 10-inch round. Scoop one-fourth of the cheese filling onto one side of the dough round and fold the other side over; press the edges firmly to seal. Transfer the khachapuri seam side down to a baking sheet and form into an elongated oval with tapered ends. Repeat with the remaining balls of dough and filling. Brush all over with the beaten egg yolks.
Using scissors, cut a 6-inch slit down the center of each khachapuri; widen the slit to expose about 2 1/2 inches of the filling. Bake for about 30 minutes, shifting and rotating the pans halfway through, until the breads are puffed and golden and the cheese is melted.
Stir the cheese filling and crack an egg in the center of each bread. Bake for 8 to 10 minutes, until the egg whites are almost set and the yolks are runny. Tuck 1 tablespoon of the butter into each khachapuri, brush the breads with melted butter and serve hot. Mix the butter and egg into the cheese filling before eating.