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 filling. Add butter, half a glass of milk, salt and pepper to the boiled potatoes and mash to make a puree.
When the dough has risen, divide it into equal 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. Wrap the filling in the dough, gathering it to the top. Press with your hands, turn over and knead lightly. Roll out carefully with a rolling pin.
Place the khachapuri in the pan, cover with a lid and fry on low heat for about 8 minutes until the dough sets on top. Turn over. Fry for another 3 minutes until the dough becomes golden brown. Place the finished flatbread on a plate and grease with a piece of butter.