Preheat oven to 350°F. Line a rimmed baking sheet with foil, then mist with cooking spray.
Warm oil in a small nonstick skillet over medium heat. Add bell pepper and onion and cook, stirring, until tender but not browned, 8 to 10 minutes. Remove vegetables to a plate to cool.
In a large bowl, gently mix chicken, ¼ cup barbecue sauce, mozzarella and onion mixture. Form into 6 patties and place on baking sheet. Bake until an instant-read thermometer inserted in center of a burger reads 160°F, 15 to 20 minutes.
In a small bowl, mix mayonnaise and remaining 2 Tbsp. barbecue sauce. Spread 2 tsp. on bottom half of each bun, top with a patty, cover with top half of bun and serve.