Using a stand mixer, combine the flour, salt, yeast and sugar in a large bowl.
Heat up water and milk in a pot until it reaches 115 degrees Fahrenheit. Once done, pour the mixture into the bowl of dry ingredients.
Using the hook attachment of the stand mixer, knead the dough until it becomes smooth and elastic.
Add the olive oil into the dough and knead for another minute.
Get a deep bowl, drizzle a little bit of olive oil onto the bottom and sides then place the kneaded dough and cover it using a plastic wrap. Let it rise for 1 hour in a dry and warm space until it doubles in size.
Uncover the dough and press into the dough a few times to release some air. Cover it again with the plastic wrap and let it sit for 30 minutes in a warm space.
In a different bowl, combine the mozzarella, farmer’s cheese and feta cheese.
Prepare a surface, sprinkle some flour and then remove the dough from the bowl and transfer it to the floured surface. Then, cut it into 4 equal parts.
Spread and shape each piece of the dough into a 9-inch diameter circle. Then, shape it like a boat by rolling the 2 opposite sides of the dough towards the centre. After shaping, pinch the corners together.
Line a baking tray with greased parchment paper and transfer the khachapuri.
Stuff the khachapuri with the cheese mixture, beat an egg, then add a teaspoon of water. Brush the dough with the egg wash.
Preheat the oven to 450 degrees Fahrenheit and bake the khachapuri for about 15 minutes or until the colour becomes golden brown.
Using the back of a spoon, make a well in the centre of the khachapuri with about 3 inches diameter then crack 1 egg into each well. You can also stick a few pieces of butter into the cheese.
Bake the khachapuri again for another 5-6 minutes (cooking time may depend on your oven). Your goal is to cook the egg white but still pretty runny.
To serve, use a fork to mix the egg and cheese. Serve the khachapuri immediately while hot.