In a large bowl, combine flour, yeast, sugar, and salt. Make a well in the center and pour in warm milk and olive oil. Mix until a shaggy dough forms.

Turn the dough out onto a lightly floured surface and knead for 8-10 minutes, or until smooth and elastic. Lightly grease a clean bowl with olive oil, place the dough in it, turning to coat. Cover with plastic wrap and let rise in a warm place for 1-1.5 hours, or until doubled in size.

Preheat your oven to 450°F (230°C). If using a pizza stone, place it in the oven while preheating. Meanwhile, combine the shredded cheese with the large egg in a bowl.

Once risen, gently punch down the dough. Divide it into two equal portions. Roll each portion into an oval shape, about 10-12 inches long and 6-8 inches wide.

Sprinkle half of the cheese mixture evenly over the center of each oval, leaving a 1-inch border. Fold the longer sides of the dough inwards, pinching the ends firmly to form a boat shape, leaving the cheese exposed in the center. Pinch the two ends of the boat together to seal.

Carefully transfer the khachapuri boats to a baking sheet lined with parchment paper (or directly onto a preheated pizza stone).

Bake for 12-15 minutes, or until the crust is golden brown and the cheese is bubbly.

Remove from the oven. Carefully crack one fresh egg yolk into the molten cheese center of each khachapuri and place 1 tablespoon of cold butter around the yolk.

Return to the oven for another 2-3 minutes, just until the egg white is set but the yolk is still runny. Do not overcook the yolk.

Serve immediately. Diners can stir the egg yolk and butter into the hot cheese before tearing off pieces of the crust to dip, enjoying the rich, creamy center.