Sift flour (530 g) in a deep bowl, add yeast, salt, sugar, room temperature milk, vegetable oil and 20 grams of softened butter. All mix thoroughly until smooth, if necessary, pour a little water (about 50 ml). Cover the bowl with dough, leave for thirty to forty minutes. Divide the dough into five servings – they should be used immediately or sent to the refrigerator.
To make the filling, you need to grate the whole cheese coarsely, mix it with the egg and add 20 grams of butter – if the cheese is greasy, you can do without the butter: the filling should be fairly loose.
Roll out the five servings of dough with a rolling pin so that you get quite large ovals 5 mm thick. Along the perimeter, make punctures with a toothpick or wooden skewer to release air.
Put 150 grams of cheese filling (portions of Imeretinsky cheese, suluguni and one egg) into portions of rolled dough, connect the edges of the dough and pinch it – they should resemble boats in the form of khachapuri.
In the middle of each cake, gently stretch the dough with your hands, make an incision about 4 cm long and tuck the edges of the dough inside the khachapuri so that the filling is visible.
Once again pierce the cakes around the perimeter, arrange them into dry baking tins and put them in an oven preheated to the maximum temperature for fifteen minutes.
4-5 minutes before readiness, you need to get the dish out of the oven, put the egg and send “reach” again.
Before serving, carefully place two small pieces of butter on the egg middle pies.