Warm the milk and water until they are warm but not hot.
Add butter, salt, yeast and sugar to the milk. Beat in the egg. Add warm water and gradually add flour. The dough should be a little sticky, don't overfill it with flour. Cover the dough and let it rise.
Punch down the dough and divide into 4 parts of approximately 220 grams each.
Grate the cheese on a coarse grater and mix.
Roll out each piece of dough to a diameter of 25 cm and sprinkle with cheese. Roll into a boat shape and bake for 12-13 minutes. Add the egg to the center and place in the oven for another 2-3 minutes.
Grease the finished khachapuri with a piece of butter, adding the remaining butter to the cheese and egg.