Grilling draws such great flavors from vegetables. So slide the meat over and give this veggie gyro a chance.
For pesto vinaigrette, in a food processor or blender combine basil, 1/2 cup of the oil, the Parmesan cheese, nuts, vinegar, and garlic. Cover and process or blend until smooth. Season to taste with salt and black pepper. Cover and set aside.
In a large bowl toss together squash, sweet peppers, the remaining 1/4 cup oil, and oregano. Thread vegetables onto 12-inch skewers,* alternating squash and pepper pieces.
For a gas or charcoal grill, place skewers on the rack of a covered grill directly over medium heat. Grill for 12 minutes or until tender, turning once halfway through grilling. Meanwhile, lightly brush cheese slices and flatbreads with the additional olive oil. Place on grill rack; cover and grill about 2 minutes or until heated through and cheese is golden brown, turning once halfway through grilling.
Fold flatbreads in half. Fill each with 1/2 cup of the lettuce, two grilled cheese slices, and the vegetables from one skewer. Top each with some of the onion and drizzle with some of the pesto vinaigrette.
If using wooden skewers, soak in water for at least 30 minutes before grilling.