In a large bowl, combine flour, yeast, sugar, and salt. Gradually add lukewarm water (or milk) and oil (or melted butter). Mix until a shaggy dough forms.

Knead the dough on a lightly floured surface for 5-7 minutes until smooth and elastic. Place in a lightly oiled bowl, cover, and let rise in a warm place for 1-1.5 hours, or until doubled in size.

While the dough rises, prepare the filling. In a separate bowl, combine the grated cheese and one large egg. Mix well.

Once the dough has risen, punch it down gently. Divide into two equal portions (for two medium khachapuri). Roll one portion into a round disc, about 8-10 inches in diameter.

Place half of the cheese filling in the center of the dough, leaving a 1-inch border. Gather the edges of the dough towards the center, pinching them together to completely enclose the filling, forming a ball.

Gently flatten the ball into a round disc again, about 7-8 inches in diameter. Be careful not to tear the dough. Repeat with the second dough portion and remaining filling.

Preheat oven to 200°C (400°F). Line a baking sheet with parchment paper. Carefully transfer the khachapuri to the baking sheet.

(Optional) Whisk the egg yolk with a teaspoon of water and brush over the top of each khachapuri for a golden finish.

Bake for 15-20 minutes, or until golden brown and puffed.

Once baked, brush immediately with melted butter. Serve hot and enjoy!