In a large bowl, whisk together the milk, sour cream, 1 tablespoon of the oil, yeast, and sugar to combine.

Switch over to a wooden spoon and gradually begin to add the flour, along with the salt, a bit at a time, mixing to incorporate after each addition.

Once the spoon becomes cumbersome to work with, switch over to your hands and knead until all the flour has been incorporated and you have a cohesive dough.

Cover with a clean kitchen towel and allow the dough to rest for 15 minutes.

Transfer the dough to a lightly greased surface. Pour a little of the remaining 1 tablespoon oil on your hands and begin to knead the oil into the dough.

Once no oil remains, continue to knead for 4 to 5 minutes.

With lightly greased hands, form the dough into a ball and transfer to an oiled bowl. Cover the bowl with a slightly damp kitchen towel and set aside to proof in a warm place for 1 hour.

Turn the dough by taking one edge and gently stretching it up and over toward the center. Repeat until you’ve stretched all the edges.

Cover again and proof for another 1 hour. Turn the dough again.

Turn the dough out onto a lightly floured surface. Pat the dough into a rectangle and cut the dough into 4 equal portions (about 200 g each).

Form into even balls and place on a generously floured surface, cover with the kitchen towel, and let rest until the balls are soft and slightly puffy, 15 to 30 minutes.

In a medium bowl, stir together the mozzarella, feta, ricotta, egg, and flour until thoroughly combined. Divide into 4 sturdy balls and set aside.

Take one piece of dough, keeping the others covered, and pat it out gently with your fingers into a 6-inch (15 cm) round with the center a little thicker than the edges.

Place one ball of filling in the middle. Carefully grab two opposite sides of the dough, bring them up over the filling, and pinch them together.

Hold them securely together with one hand as you bring up the remaining edges over the filling, pleating as you go and pinching the ends together tightly at the top to secure like a drawstring purse.

Gently start flattening out the filled dough with your fingers. Sprinkle a little flour and flip it over.

Continue to flatten the dough with your fingers and palms (you can also use a rolling pin), gently working from the center out to the edges to spread the filling out evenly, until you have a 9- to 10-inch (23 to 25 cm) round.

Heat a 12-inch (30 cm) nonstick skillet over medium heat. Once hot, carefully transfer the khachapuri to the pan.

Cook until the bottom begins to firm up and brown spots develop all over, 4 to 5 minutes.

Carefully flip over and glide the stick of butter over the top like a glue stick to coat the khachapuri evenly.

Cook for another few minutes until the bottom is browned—if the khachapuri begins to inflate, pierce it with the end of a sharp knife to release steam.

Flip the khachapuri over onto a serving plate—the unbuttered side should be facing up—and give the second side a light coating of butter.

Repeat the process with the remaining pieces of dough, stacking the khachapuri on top of each other as they come out of the pan.

Let cool a bit for the cheese to set before cutting the stack into eighths. Serve immediately.