To make dough:

Warm milk to lukewarm temperature (not boiling). Sprinkle yeast and sugar over milk and stir to dissolve. Set aside to proof for 5-10 minutes – mixture will become bubbly and foamy.

Meanwhile place 2 cups bread flour and salt in a large mixing bowl. Whisk to blend; make a well in the center, pushing flour against sides of bowl.

Add proofed yeast mixture and butter to the well. Mix flour slowly until a dough forms and comes away from sides of bowl. Add softened butter, kneading it into the dough.

Knead with your hands until dough is a soft and elastic pliable ball, adding in remaining ¼ cup flour as needed if dough is too sticky.

Cover bowl with a towel and allow to rise until dough is doubled in bulk, about an hour.

To Shape, Fill & Bake:

Preheat oven to 425°F.

Combine all ingredients in a large mixing bowl, tossing until well mixed. Set aside.

Divide dough in half. Place one half of dough on a floured board or parchment paper. 

Using a floured rolling pin, roll out into an oblong shape about a ¼” thick. 

Mound half the cheese mixture in the middle of the dough, pressing and spreading into an even layer, leaving a border of dough plain. 

Fold the long sides over the dough on each side. Twist and seal the ends together on each side to form a boat shape. 

Brush folded dough with egg wash. 

Bake for 10-14 minutes, or until dough is lightly browned.

Remove from oven; using the back of a spoon, make an indentation in the center of the cheese. Pour a whole cracked egg into the indentation. 

Bake for an additional 5-7 minutes or until egg white is mostly set.

Remove from oven and add 2-3 tablespoons butter to the middle of the khachapuri. 

Use a fork to mix up the egg and butter into the cheese filling. Serve immediately. 

Repeat with remaining dough and cheese.