Combine the lukewarm water, yeast, and sugar. If using active dry yeast, let it stand for 5–10 minutes until foamy; instant yeast may be mixed directly with the flour.

Mix the flour and salt in a large bowl, then add the yeast mixture and oil. Knead for 8–10 minutes until smooth and elastic, keeping the dough soft and slightly tacky rather than wet.

Place the dough in an oiled bowl, cover it, and let it rise in a warm place for 60–90 minutes, or until doubled in size. Meanwhile, combine the grated mozzarella and crumbled feta, keeping the cheese mixture fairly dry.

Heat the oven to 230°C/450°F with a baking stone, steel, or heavy baking sheet inside. Divide the risen dough into 4 equal pieces and roll each into an oval about 25–30 cm long.

Roll or fold the long edges inward and pinch the two ends together to create a boat shape with an open center. Transfer the boats to parchment paper and fill each with one-quarter of the cheese mixture.

Bake for 12–15 minutes, until the crust is deeply golden and the cheese is bubbling. Remove the tray briefly, make a shallow well in each cheese center, and crack one egg into each well.

Return the khachapuri to the oven for 2–4 minutes, just until the egg white is set while the yolk remains soft. Add a piece of butter to each hot bread and serve immediately.