Working with one ball of dough at a time, stretch dough with floured hands or a rolling pin into an oblong or circle roughly 10 inches long and 6 inches wide. Once you have 2 or 3 pieces stretched, lay them out directly above the hot side of the grill. Cook without moving, popping any large air bubbles with a knife or the edge of a spatula or pizza peel, until bottom side is charred in spots and light golden brown. Flip with a large spatula, pizza peel, or tongs and cook until other side is charred and browned. Remove bread from the grill, transfer to a large plate, and cover with a clean dish towel to keep warm while you cook the remaining bread.
To Serve: Form the za’atar mixture by combining oregano, thyme, savory, sesame seeds, sumac, and salt in a small bowl and stirring with fingers until homogenous. Brush finished flatbreads with olive oil and sprinkle generously with the za’atar mixture. Place labne in a serving dish, drizzle with olive oil, and sprinkle with za’atar. Serve bread with labne immediately.
