In the bowl of a stand mixer with the dough hook attached, combine the lukewarm water and milk. Add in yeast, sugar, and salt. Stir to combine. Let it rest for a few minutes or until it is bubbly.
In small parts, add flour to the liquid mixture. Knead on low speed as you add in flour. Add oil to the mixture. Continue kneading until the dough is elastic and no longer sticky.
Cover the dough with a towel and let it rise for about an hour.
Khachapuri Filling
In a large bowl, combine together the farmers cheese, feta, and Mozzarella until the mixture is creamy and smooth.
Divide the dough into 6 equal parts. Roll out each one into a 7 inch round shape.
Roll up two ends and pinch the edges, leaving the center open for the filling. The dough should resemble a boat shape.
Place the cheese filling in the center. Leave them to rise for about 30 minutes covered with a towel.
Bake at 425°F for 15 minutes or until lightly golden.
Cut a slit or make an indent in the center of cheese filling, add an egg to each opening and 1/2 tbsp of butter. Return to the oven for 5 more minutes. Serve immediately.