Warm the milk until lukewarm. Mix it with the yeast and sugar in a bowl and let stand for about 5 minutes, until lightly foamy. Add the yoghurt, olive oil, salt and flour. Knead into a soft, smooth dough.
Let the dough rise
Cover the dough and let it rise at room temperature for about 1 hour, until doubled in size.
Prepare the filling
Mix the grated mozzarella and crumbled feta in a bowl.
Shape the khachapuri
Roll the dough into a round disc. Place the cheese filling in the centre, then bring the edges together and seal tightly.
Flatten and bake
Gently flatten into a thick round. Bake at 250 °C (480 °F) for 12–15 minutes, until golden and cooked through.
Serve
Let rest briefly, then slice and serve hot.
Notes
Open khachapuri with egg (Adjaruli style)
Instead of sealing the dough, roll it into an oval, add cheese to the centre, fold the sides inward and pinch the ends to form a boat shape. Bake as above, then crack an egg into the centre, add a knob of butter, and return to the oven for 1–2 minutes until the egg white sets.