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1
Make the dough: In the bowl of an electric mixer fitted with the dough hook attachment, mix the flour, yeast, sugar, and salt to combine. Add the butter and mix on low speed until the mixture looks crumbly, about 1 minute.
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2
Add the milk and water and mix the dough on low speed for 4 minutes. Raise the speed to medium and mix until the dough is very smooth, 2-3 minutes more.
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3
Transfer the dough to a large, oiled bowl and cover with plastic wrap. Let the dough rise for 1 1/2 to 2 hours, until the dough is puffed (it may not fully double in size).
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4
Divide the dough as desired: Cut into two even pieces for larger, more share-able khachapuri, or into four even pieces for smaller, individual khachapuri.
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5
Pat the dough into an oval shape with your hands. On a lightly floured surface, roll out the dough, maintaining the oval shape, into an oblong oval about 1/3-inch thick.
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6
Pick the dough up from the ends and transfer it to a parchment lined baking sheet. Gently stretch the dough when you transfer it to help elongate the oval a bit.
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7
Crumble the goat cheese evenly over the dough, leaving about 1 inch around the edge on all sides. Sprinkle the mozzarella evenly over the goat cheese.
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8
Working your way around the bread, fold the excess dough up and over the filling to create a little wall and encase it. Gently pinch the ends of the oval to help seal. Cover the breads with a piece of plastic wrap sprayed with nonstick spray, and let the bread rise for 30 minutes to 1 hour.
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9
Preheat the oven to 375u00b0 F. Brush the edges of the dough with egg wash.
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10
For small khachapuri, crack an egg on each piece, in the center of the bread. Transfer to the oven and bake until the crust is golden brown, the cheese is melted and lightly golden, and the egg white is set, but the yolk is still runny, 30 to 35 minutes.
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11
For large khachapuri, transfer the breads to the oven and bake for 15 to 20 minutes. Remove the breads from the oven and crack three eggs evenly over the center of each bread. Return to the oven and continue to bake for 25 to 30 minutes more, until the bread is golden, the cheese is melted, and the egg white is set but the yolks are still runny.
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12
Let the khachapuri cool for 5 to 10 minutes before serving warm.
