Prepare the dough starter. Warm the water slightly, add 2 teaspoons of sugar, and mix well. Add dry yeast, cover with plastic wrap or a towel, and let sit for 30 minutes.
Slightly warm the milk, melt the butter in the microwave, and let it cool. Sift the flour with salt into a large bowl, then add the egg, milk, and melted butter.
Pour in the yeast mixture and knead the dough. Knead for 5-10 minutes until elastic. Coat the dough with vegetable oil, place it in a deep bowl, cover, and let rise for 2 hours.
Divide the dough into 3 parts. Roll one portion of dough into a circle, place about 150 g (1 ¼ cups) of grated cheese in the center. Fold the edges of the dough and gather them in the center.
Flip the dough ball seam-side down and roll it into a circle again. The dough will have air inside, which you can release by poking holes in the khachapuri. Brush the top with a beaten egg.
Preheat the oven to 200° C/400° F. Transfer the khachapuri to a baking sheet and generously sprinkle it with cheese. Bake for 8-10 minutes, then increase the temperature to the maximum and bake until golden brown (approximately 5 minutes).
Serve warm—perfect with tea or on its own.