Combine the lemon juice, salt, oil, spices, and cilantro in a large bowl. Add the chicken and toss to coat.
Preheat the outdoor grill to medium-high heat.
Grill the chicken 4 to 5 minutes on each side, or until cooked through. Cut into bite- size pieces.
Meanwhile, to make the yogurt and garlic sauce, combine the garlic, salt, and lemon juice in a blender until smooth. Gradually add the oils in a thin, steady stream while the motor is still running, until the sauce has thickened; fold in the yogurt and sumac.
To make the salad, combine the parsley, tomato, and onion.
To assemble, place the flatbreads on the grill for about 5 to 10 seconds to warm slightly, then spread the yogurt and garlic sauce over them. Arrange the tomato-parsley salad, lettuce, and chicken on top. Roll up to enclose and cut in half to serve.