In a large mixing bowl, combine the flour, yeast, sugar, salt, vegetable oil, water, and milk. Mix well and knead the mixture by hand into a smooth dough.
Cover the bowl with plastic wrap, poke a small hole in the center to let it breathe, and leave it to rise. Once it has risen, punch the dough down and let it rise for a second time.
Prepare the filling by grating the Suluguni and Imeretian cheeses, or passing them through a meat grinder. Mix the two cheeses together evenly.
For each khachapuri, take 180 grams of the proofed dough and roll it out into an oblong shape. Place 200 grams of the cheese mixture in the center and tightly pinch the edges of the dough closed around the cheese to form a sealed pocket.
Brush the top of the dough pocket with a beaten egg, then flip it over and press it into a layer of sesame seeds to thoroughly coat the surface.
Place the coated dough on a baking sheet or pan. Using scissors or a knife, carefully slice down the center of the dough and peel the edges back to create an open boat shape, exposing the cheese filling.
Bake in an oven preheated to 220°C (430°F) for 7 to 10 minutes, until the dough is cooked through and the cheese is golden and bubbly.
Remove the khachapuri from the oven. Immediately place a pat of butter and a raw egg yolk into the center of the hot cheese. Stir the egg and butter into the melted cheese right before eating.
Cook it hands-free.
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