Prepare the dough. Mix flour with butter. Separately, beat eggs, vinegar and salt into cold water and mix. Add liquid to flour and mix. Do not knead the dough, but lift it from different sides and press it with your hands to the center, creating layers until it becomes smooth. This will take about 15 minutes. Wrap in film and put in the refrigerator for 2 hours.
While the dough is resting, prepare the cheese filling. In a bowl, mix the grated cheese, egg and butter. Set aside on the table.
Prepare the khachapuri for baking. Roll out the dough thinly on a floured surface and divide into squares. Place the filling in the center of each square, avoiding the edges of the dough. Fold the corners of the dough to form a square and press it to seal. Prick the dough in several places with a fork to release air.
Place the finished pies on a baking sheet with the seams down and bake in the oven at 200 degrees for 20 minutes.
Serve khachapuri on one large plate. Individual serving is not accepted in Georgia. All dishes are placed on a common table, if necessary, they are divided into portions.