Combine flour, salt, yeast, and sugar in a bowl. Then add warm milk. Using the hook attachment, knead the dough until it takes shape. Then add oil and knead some more.
Place the dough in a greased bowl. Cover and let it double in size. Remove from bowl and divide the dough into 4 boat-shaped pieces. Place the bread boast on a baking sheet lined with parchment.
Combine the three kinds of cheese and fill each bread boat. Using a pastry brush, brush with egg wash and bake for 15 minutes at 450 degrees Fahrenheit.
Pull the khachapuri out of the oven, then make a well in the middle. Drop the egg in the center, add pieces of butter to the melted cheese, and bake for another 5-6 minutes.