Prepare the dough. Heat the milk until it is slightly warm and dissolve the yeast in it. Pour 2.5 cups of flour into a deep bowl. You will need the remaining half cup later. Pour the milk into the flour. Add the egg and butter. Mix thoroughly. The dough should become smooth and soft. Cover with cling film and put in a warm place for 60 minutes.
While the dough is rising, prepare the cheese filling. In a bowl, mix the grated cheese, egg and butter. Leave on the table.
Prepare the khachapuri for baking. Mix the risen dough with the remaining flour, knead and divide into 3 parts. Roll each part with a rolling pin into round flat cakes about 0.5 cm thick.
Place the filling in the center of each flatbread, dividing it into 3 equal parts. Wrap the filling in the dough, gathering it to the top. Compact with your hands, turn over and knead lightly. Roll out carefully with a rolling pin. Pierce with a fork in several places to allow air to escape during baking.
Place on a baking tray and place in an oven preheated to 180°C for 15 minutes.