1. To make the dough: Place the yogurt and egg in the bowl of an electric mixer fit with the paddle attachment. Add about 2/3 of the flour with the salt and baking powder, and mix on medium speed until homogeneous.
2. Add the butter and the rest of the flour, and mix again. Remove the bowl from the mixer, and knead the dough by hand for about one minute. Cover with plastic wrap, and refrigerate for 30 minutes.
3. To make the khachapuri: Divide the dough into 2 equal balls.
4. Preheat the oven to 450° F, and place a dish full of water on the bottom rack.
5. Slice the mozzarella and feta. In a bowl, crumble both cheeses between your fingers.
6. Season with black pepper, and adjust the salt level if necessary—the mixture should taste quite salty, but still be edible!
7. On a floured surface, roll each ball of dough into an 8-inch diameter disc (a perfect shape isn’t very important at this point). Place the cheese mixture in the center, then wrap the dough around it, just like you would wrap something in a handkerchief. Flip the dough over, and gently roll each khachapuri back into an 8-inch diameter disc. Alternately, you can wrap the dough around the cheese mixture like a purse, cut the top, and roll each khachapuri without flipping it. (It’s OK to keep some cheese uncovered in the center.)