Whisk 7 grams of active dry yeast and a tablespoon of sugar into a cup of warm whole milk and set aside for 10 minutes
In a large bowl, add 2 1/2 cups of bread flour, a teaspoon of salt and 1/3 cup of olive oil
Once the yeast has bloomed and the mixture is foamy, add this to the flour mixture and mix until all of the dry flour has been incorporated
Knead the dough for about 5 minutes and then form the dough into a round ball and transfer into a large greased bowl
Cover the bowl with a towel and let the dough sit in a warm place for about 25 minutes
While dough is resting, combine 7 ounces of soft farmers cheese with 8 ounces of shredded mozzarella, 1 large egg and a pinch of salt and black pepper and set aside for now
Divide the dough in half and roll each portion out to about a 12 inch long oval
Form the portions into a boat like shape by rolling the long sides in 2-3 times and bringing the ends in towards themselves pinching them together to close (see photos and further instructions above)
Transfer the shaped dough onto a sheet of parchment paper and divide the cheese filling between the two portions
Cover the khachapuri and let it sit in a warm place for 25 minutes
Preheat your oven to 350 degrees
Whisk an egg in a small dish and brush the exposed edges of the dough (set the leftover egg aside to use later)
Bake the khachapuri in a 350 degree oven for 15 minutes and then transfer the baking sheet out of the oven onto a heat proof surface (close the oven door quickly to maintain the interior temperature)
Use the back of a spoon to form a dip in the middle of the melted cheese and crack an egg into this spot
Brush the dough with the remaining whisked egg and return the khachapuri to a 350 degree oven for another 4-8 minutes (depending on how set you prefer your egg)
Once the egg is done to your liking, transfer the khachapuri onto plates and top with a tablespoon of butter
Use a fork to mix the melted cheese, egg and melting butter together and enjoy!