In a bowl, mix the warm water, sugar, and yeast, and let sit until foamy, about 5 minutes.

Stir in the yogurt and olive oil, then gradually mix in the flour and salt to form a soft dough.

Knead the dough for about 8 minutes until smooth and elastic, then cover and let rise for 1 hour, or until doubled.

Divide the dough into two portions and roll each into an oval about half an inch thick.

Fill the center of each oval with shredded cheese, then roll and pinch the long edges up into tall points, twisting the ends to form a boat shape.

Fill the center of each boat generously with more shredded cheese and bake in a preheated 475°F oven for about 12 to 15 minutes, until the crust is deep golden.

Remove from the oven, crack one egg into the center of each hot cheese pool, and return to the oven for 2 to 3 minutes, just until the egg white sets but the yolk stays runny.

Top each khachapuri with a pat of butter and serve immediately, stirring the egg and cheese together at the table before tearing and dipping the crust.