Place the yoghurt 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.
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.
Divide the dough into two.
Preheat oven to 225°C (437°F), and place a dish full of water on the bottom rack.
Slice the mozzarella and feta. In a bowl, crumble both cheeses between your fingers. Season with black pepper, and adjust the salt level if necessary — the mixture should taste quite salty, but still be edible!
On a floured surface, roll each ball of dough into a 20 cm large disc (it is not necessary to make a perfect round shape). Place the cheese mixture in the center, then wrap the dough around it, like you would wrap something in a handkerchief. Flip the dough over, and gently roll each khachapuri back into a 20 cm large disc.
Transfer the khachapuri to a baking sheet lined with a parchment paper. Brush the dough with a mix of egg yolk.
Bake in the oven for 18-20 minutes, until the top is golden brown. Let cool 5 minutes, then slice and serve.