In a pre-heated oven place a full sheet pan. Meanwhile toss vegetables and fruit (mushrooms, peppers, onion, pineapple, squash, zucchini, celery, and carrots) in oil and roast on screaming hot sheet pan until golden brown, approximately 15-20 minutes.
While veggies roast, make the sauce - whisk remaining ingredients in a sauce pan over medium heat. When mixture comes to a boil reduce heat and add roasted veggies, remove from heat. Serve on hamburger buns with 2 slices of cheese per sandwich. Serve warm.