Drain the mozzarella and coarsely grate it on a grater. Place grated mozzarella on a colander to let excess liquid drain. Crumble the feta and mix together with the mozzarella.
After 2 hours have passed, transfer the dough to a floured surface and divide it into 2 balls. Roll them out with a rolling pin trying to get a long oval shape on both. Transfer rolled out dough to an oven tray lined with baking paper.
Divide filling between two pieces of dough, leaving about 1-2 cm from the edges. Fold the side edges over the filling on both sides, then press ends together and give a slight twist to securely close them. This will prevent the dough from snapping open as it bakes, losing the boat shape and letting the filling pour out.
Bake in the preheated oven at 250°C (480°F) for 12 minutes. During this time, the cheese should melt completely and the dough should get a golden tone.
After 12 minutes, take khachapuri out of the oven and crack an egg in the middle of each filling. Divide room temperature butter between the 2 fillings, then return the breads to the oven for 3 more minutes. The egg white should be cooked enough to lose its translucent appearance, but the yolk should still be runny when the bread is served.