Mix all-purpose flour, yeast, lukewarm water, milk (if using), sugar, and salt in a mixing bowl. Knead the dough for 8-10 minutes until smooth and slightly tacky but not sticky. Let dough rest for 10 minutes to relax gluten before proofing.
3 cups all-purpose flour, 2 teaspoons active dry yeast, 1 cup lukewarm water, ¼ cup milk, 1 tablespoon sugar, 1 teaspoon salt
Cover the dough with a damp towel and proof in a warm spot (about 75°F) for 1 to 1.5 hours until doubled in size. Meanwhile, preheat your oven to 450°F with a pizza stone or baking sheet inside for 30 minutes.
Divide the proofed dough into 4 equal pieces. Roll each piece into a boat shape, folding the edges about ½ inch thick to create a sturdy rim while thinning the center for the cheese filling. Pinch the pointed ends tightly to prevent leaking.
Fill each shaped dough boat with grated suluguni cheese. Bake in the preheated oven for 8 minutes until the edges start to brown and the cheese begins melting.
8 ounces suluguni cheese
Remove breads from oven and crack one room temperature egg into the center of each boat. Return to oven and bake for 4 to 7 minutes until egg whites are set but yolks stay runny. Check at 4 minutes to avoid overcooking.
4 eggs
Brush warm khachapuri immediately with melted butter for a glossy, rich crust. Serve hot to preserve creamy cheese and runny yolk.
2 tablespoons butter
