In a large bowl, combine flour, yeast, sugar, and salt. Add warm water and olive oil. Mix until a shaggy dough forms. Knead on a lightly floured surface for 5-7 minutes until smooth and elastic.

Place the dough in an oiled bowl, cover with plastic wrap or a clean towel, and let rise in a warm place for 1-1.5 hours, or until doubled in size.

While the dough is rising, prepare the filling. In a medium bowl, mix the grated cheese with one large egg until well combined.

Once the dough has risen, punch it down gently and divide it into 2-4 equal pieces. On a lightly floured surface, roll each piece into an oval shape, about 8-10 inches long.

Fold the long edges of each oval inward towards the center, creating a boat shape. Pinch the ends firmly to seal them, forming a crust around the edges.

Spoon the cheese filling into the center of each dough boat.

Transfer the filled Khachapuri boats to a baking sheet lined with parchment paper. Cover lightly and let rise again for 15-20 minutes while you preheat the oven.

Preheat your oven to 450°F (230°C).

Bake the Khachapuri for 10-12 minutes, or until the crust is golden brown and the cheese is bubbly.

Remove the Khachapuri from the oven. Gently make a well in the center of the hot cheese in each boat and crack one egg into each well. Return to the oven for another 3-5 minutes, or until the egg whites are set but the yolk is still runny.

Remove from the oven and immediately place a pat of butter (about 1 tbsp) next to the egg yolk in each Khachapuri. Serve hot. Encourage diners to swirl the egg yolk and butter into the hot cheese before eating.