Thaw the puff pastry, then sprinkle the board with flour and roll it out into a layer, not very thin. Cut an oval shape from the layer.
Grate the cheese and place it on the dough in the middle, leaving the sides. Carefully roll the layer on the sides into a roll, leaving the cheese untouched, to create a boat-shaped shape.
Break the egg in a bowl and beat with a fork. Brush the edges of the dough with egg and bake for 20 minutes at 180 degrees.
Remove the khachapuri from the oven, break the egg into the middle, directly onto the cheese and place pieces of butter around the edges.
Serve the dish hot with tea, coffee or compote. You can decorate with greenery.