Preheat the oven to 425 degrees F with 2 racks set in the middle of the oven.
Brush the lavash sheets on both sides with 3 tablespoons of the oil. Season lightly with salt and pepper. Cut into 2- by 1-inch rectangles. Place the rectangles on 2 baking sheets in a single layer. Bake until lightly browned and crispy, about 5 minutes. Halfway through cooking, rotate the baking sheets front to back and top to bottom, flipping the lavash pieces over if needed.
Heat a large grill pan over medium heat. Brush the chicken with 2 tablespoons of the oil and sprinkle liberally with salt and pepper. Grill until slightly charred on both sides and the internal temperature of the chicken reaches 160 degrees F, 15 to 18 minutes. Rest the chicken on a cutting board for 5 minutes.
Meanwhile, make the dressing: Whisk the honey, lemon zest, lemon juice, garlic and sumac together in a large bowl. Slowly whisk in the remaining oil. Season with salt and pepper.
To serve, slice the chicken thinly against the grain. Use gloved hands to toss the sliced chicken, romaine, cucumbers, tomatoes, mint and lavash with the dressing. Gently break up the lavash pieces with your hands to better incorporate them into the salad. Season with salt and pepper if needed. Serve family-style, or divide the salad among 4 large plates. Drizzle with additional oil and sprinkle with sumac.