There are many variations of the Georgian flatbread khachapuri, but the Adjarian style is by far the most popular. Best eaten hot, the cheese still molten from the oven, an egg yolk is cracked into it, some pats of butter added, and after a stir to mix it all into a cheesy, eggy, buttery sauce, diners tear off hunks and dip it into the filling. By no means do you have to grill this, but if you enjoy high-wire acts of outdoor cookery, grilling the flatbread is ridiculously fun.
While I tested this with a fire-brick rigged PK grill (an admittedly specific test), you might want to try using a pizza stone on your grill or simply baking it in your oven. Because the cooking surface of the fire bricks in my test is so small, you’ll need to divide your dough into two pieces and cook mini-khachapuris. (If you want to try it on a different grill or in the oven, the conventional thinking on cook times and temperature is 15 to 20 minutes in a 450° to 500° oven. Grill to grill, stone to stone, times and results will vary.)
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.