In a large bowl, combine flour, yeast, sugar, and salt. Mix well.

Gradually add lukewarm water and vegetable oil. Knead the dough on a lightly floured surface for 7-10 minutes until smooth and elastic. Cover the bowl with a clean cloth and let it rise in a warm place for 60-90 minutes, or until doubled in size.

While the dough rises, prepare the filling. In a separate bowl, combine the grated cheese and beaten egg. Mix thoroughly.

Once the dough has risen, punch it down gently and divide it into two equal balls. On a floured surface, roll out one dough ball into a circle about 8-10 inches in diameter.

Place the cheese filling in the center of the rolled-out dough, leaving a 1-inch border. Gather the edges of the dough over the filling, pinching them together at the top to seal completely, forming a ball shape.

Gently flatten the sealed dough ball with your hands or a rolling pin to create a flat, round disk, about 7-8 inches in diameter. Be careful not to break the dough and let the cheese escape. Create a small hole (about 1/2 inch) in the center for steam to escape during baking.

Repeat with the second dough ball and remaining filling.

Preheat your oven to 200°C (400°F). Line a baking sheet with parchment paper.

Whisk the egg yolk with milk to create an egg wash. Brush the tops of the khachapuri generously with the egg wash.

Bake for 15-20 minutes, or until golden brown and the cheese is bubbly. Serve hot.