Mix together dry ingredients, add in warm water and milk, combine until kneadable. Knead for 5 minutes.
Cover dough and allow to rise for two hours in the refrigerator
Once the dough has risen, roll out into a rectangular shape roughly 2 1/2 feet by 1 foot.
Begin shaping the dough into a 'boat' - to do this, roll the wide sides in toward the middle and then pinch the ends together. After that, gently spread open the rolls and begin shaping the boat.
Add the imeruli, sulguni, and matsoni into the boat and mix lightly with your fingers.
Heat your oven to 392F/200C and bake in the middle for 10-12 minutes, rotating halfway through.
Take your khachapuri out of the oven and gently add the egg yolk to the center - bake again for 1-2 minutes.
Once finished baking, add a tablespoon of butter on top, and serve hot.