Mix together dry ingredients, add in warm water and milk, combine until kneadable. Knead for 5 minutes and add flour as needed. Use the spare flour a tbsp at a time until you get a kneadable and smooth dough.
Cover dough and allow to rise for two hours in the refrigerator.
Once the dough has risen, divide in two. Reserve the second half for the second khachapuri. Roll out into a soft rectangular shape. Heat your oven to 392F/200C.
Grate sulguni and imereti cheese. see notes for substitutions.
add 1/4 cup sulguni approx 1 inch below top and bottom of rectangle. See photos above.
Fold the top edge of the dough over the cheese. Repeat with bottom. Seal both folds.
Begin shaping the dough into a 'boat' - to do this, roll the top and bottom in toward the middle and then pinch the left/right ends together. After that, gently spread open the rolls and begin shaping the boat.
Add 1 1/2 cup imeruli cheese, 1 tbsp whole milk, 1 1/2 tbsp matsoni, 1 tbsp butter evenly to the boat. Top with sulguni.
Bake in the center of the oven for 10-12 minutes. Divide one yolk and white from an egg. Remove from oven, brush with egg white and carefully add yolk to the center. Bake again for 1-2 minutes on the top rack.
Roll out remaining half of dough and repeat steps 3-9.
Once finished baking, add a tablespoon of butter on top, and serve hot.