You need more grilled bread in your life, and this grilled flatbread with za'atar is a great place to start.
This is not an authentic Middle Eastern flatbread recipe by any means. Most flatbreads from that region are lean doughs baked on a hearth. My version is hydrated, enriched with yogurt, and grilled directly over a flame (though you can also make the bread on a baking stone or steel in a hot oven). It's based mostly on my Grilled Naan recipe, which is itself not really like a traditional naan recipe at all. But no matter.
This dough is extremely forgiving stuff. In place of the water you'd use in a typical lean dough, I use yogurt or whole milk, which helps in two ways. First, the added fat limits gluten development, which means the bread stays tender even if you slightly overcook it or if your grill is not quite hot enough (I add some olive oil to this version for flavor and for even more tenderness insurance). Meanwhile, the extra milk proteins in the yogurt or milk ensure that the bread browns and chars in record time. The tanginess of yogurt also adds enough flavor that even with a brief two-hour pre-ferment, followed by a two-hour rise after dividing the dough into balls, it's still incredibly flavorful in the end. It's one of my favorite dough recipes, and the one I turn to when I know I want to make bread and eat it in the same afternoon.
The dough is really adaptable, too. I've successfully cooked flatbreads using this exact dough on a charcoal grill, a gas grill, in a gas-fired pizza oven, on a baking steel in a regular oven, in a cast iron skillet on the stovetop, and in an honest-to-goodness tandoor oven! You might notice that the bread is extra crisp and a little stiff right off the grill. That's okay. Think of it as a corn tortilla: You need to stack the breads in a clean towel and let them steam together for a moment or two so they can fully soften to their final, perfect texture.
The real key here is not to skimp on the za'atar. This is not a light dusting or even a big pinch. This is bread pretty much caked in the stuff, and that's how it should be. The bread is really a vehicle to deliver the za'atar to your mouth, and if you happen to have some good-quality labne lying around, or even some nice thick yogurt, make sure to sprinkle more za'atar on top of that before serving it on the side with a drizzle of olive oil.