Whisk to combine the flour, yeast, sugar and salt in stand mixer bowl.
Pour warm milk and 3/4 of water into the bowl of dry ingredients.
Mix the ingredients with the hook attachment, slowly adding more water until it comes together, about 2 to 3 minutes. You might not need to use all the water.
Mix until dough pulls from the sides and looks smooth, about 5 to 7 minutes.
Add the olive oil into the dough and knead for another minute.
Remove from bowl and form into a round. Place in lightly greased bowl. Cover and place in a warm place to proof for about an hour or until dough doubles in size.
Deflate air from dough. Reshape into a round. Cover and let it rest for another 30 minutes.
While waiting, combine the cheese and set aside.
Preheat oven to 450°F (232°C).
Remove the dough from the bowl. Cut it into 4 equal pieces.
Roll out into approx. 10” (25cm) in diameter rounds.
Roll 2 opposite sides of the circle towards the center, creating an oval, boat-like shape. (The rolled edges will keep the melting cheese securely contained). Pinch the corners together to seal.
Transfer the khachapuri onto a silicone or parchment lined baking sheet.
Carefully spoon the cheese onto each of the shaped dough.
Brush egg wash on the dough (Beat 1 egg with a teaspoon of water.)
Bake in a preheated oven for about 15 minutes.
Remove from oven and quickly make a deep well in the center of each khachapuri with the back of a large spoon and drop 1 egg into each well. Optional, stick a few small slices of butter into the cheese.
Pop the the khachapuri back into the oven and bake for another 5-6 minutes (or longer depending on your oven) At this point, the egg white should be white, and still pretty runny. You want the yolk to be jammy, and avoid over cooking.
Remove from oven, top with your favorite herbs (I used green onions), mix the cheese and runny egg and serve promptly.