Preheat the oven to 425 degrees F. Toss the mushrooms, onion, half of the tomatoes, 1 tablespoon olive oil, the rosemary, 1/2 teaspoon salt and a few grinds of pepper on a rimmed baking sheet. Roast until the vegetables are tender, about 15 minutes. Slice the mushrooms.
Meanwhile, combine half of the chickpeas in a medium bowl with 1 tablespoon each olive oil and balsamic vinegar and smash with a fork. Season with salt and pepper. Spread on the lavash and top with the muenster and parmesan.
Drain any liquid from the vegetables; arrange lengthwise down the center of the lavash. Top each with 1 cup arugula, then roll up, starting with a long side. Arrange on a baking sheet; bake until toasted and the cheese melts, 5 minutes.
Whisk the remaining 1 1/2 tablespoons olive oil and 1 tablespoon balsamic vinegar in a large bowl. Add the remaining tomatoes, chickpeas and arugula; season with salt and pepper and toss. Slice the wraps into quarters. Serve with the salad.