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1
Make the dough In a large bowl, whisk the yeast with the sugar and 1 cup of warm water.
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2
Let stand until foamy, about 10 minutes.
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3
In a small bowl, whisk the milk with the egg and 3 tablespoons of oil.
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4
Add the milk mixture, the flour and salt to the foamy yeast and mix until a dough forms.
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5
Form the dough into a ball and transfer to a lightly oiled large bowl.
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6
Cover the bowl with plastic wrap and let the dough rise in a warm place until doubled in bulk, about 1 1/2 hours.
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7
Line 2 baking sheets with parchment paper.
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8
On a lightly floured work surface, divide the dough into 4 equal pieces and form into smooth balls.
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9
Set 2 balls of dough 5 inches apart on each sheet.
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10
Cover loosely with plastic wrap and let rest in a warm place for 45 minutes.
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11
Make the filling In a blender, puree the feta with 2 of the eggs, 2 tablespoons of water and the salt.
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12
Transfer the mixture to a bowl and stir in the mozzarella.
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13
Preheat the oven to 450.
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14
Transfer the dough balls to a lightly floured work surface.
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15
Line the baking sheets with clean parchment.
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16
Roll out 1 ball of dough to a 10-inch round.
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17
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.
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18
Transfer the khachapuri seam side down to a baking sheet and form into an elongated oval with tapered ends.
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19
Repeat with the remaining balls of dough and filling.
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20
Brush all over with the beaten egg yolks.
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21
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.
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22
Bake for about 30 minutes, shifting and rotating the pans halfway through, until the breads are puffed and golden and the cheese is melted.
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23
Stir the cheese filling and crack an egg in the center of each bread.
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24
Bake for 8 to 10 minutes, until the egg whites are almost set and the yolks are runny.
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25
Tuck 1 tablespoon of the butter into each khachapuri, brush the breads with melted butter and serve hot.
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26
Mix the butter and egg into the cheese filling before eating.
