In a PK Grill, stack 8 fire bricks on their edges on the left side of the grill. Load two charcoal chimneys with charcoal heat according to instructions. Load the right side of the PK Grill with red hot coals. Close lid. Open vents. Let grill heat for 30 minutes.
Meanwhile, in a medium bowl mix mozzarella and feta and stir to combine.
Divide dough into two equal pieces. On parchment paper dusted with flour, roll one piece of dough out into a rectangle. Fold and pinch either ends to make a football shape. Roll edges in and over to make a deep reservoir in the center. Fill center with cheese mixture. Repeat with second piece of dough.
Sprinkle a handful of cornmeal on a pizza peel or overturned cookie sheet. Carefully transfer one khachapuri onto pizza peel or cookie sheet.
Slide khachapuri onto hot bricks. Close lid and back for 10 minutes. Rotate khachapuri 180° and bake 8 minutes more. Pour yolk into center of khachapuri and bake until almost set, 2-4 minutes more. Place on serving platter or cutting board, add butter, stir to combine, and serve immediately. Repeat with second khachapuri, adding more hot coals to grill if necessary.