Mince 2 red bell peppers and reserve remaining peppers for another use.
In a large bowl, add peppers, chicken, cheese, green onions, garlic, egg, and panko. Season with salt and pepper and, using gloves or a large spoon or spatula, mix ingredients until blended. Let sit for 10 minutes at room temperature. Form into 4 flat patties.
Warm a grill or grill pan over medium-high heat. If using a grill pan, add oil. Cook chicken burgers until browned on first side, about 10 minutes. Flip, reduce heat to medium, and cook other side until browned, 12-15 minutes or until an internal thermometer inserted into burger reads 165°. Transfer burgers to a clean plate to rest for 5-10 minutes.
Add buns to pan and cook until toasted to desired doneness.
Serve chicken burgers on buns with mayonnaise, lettuce, and avocado, if using.
In a large bowl, combine all ingredients except for oil. Mix thoroughly to make sure there is an even distribution of ingredients throughout burger. The egg is optional and used to help meat from falling apart on grill. Shape into 4 equal patties. Shape them a bit larger than buns as meat shrinks a bit when cooked. Set on a plate and let come to room temp for between 30 minutes to 1 hour.
Preheat grill over medium-high heat. Once hot, oil grill with a towel that has been dipped in oil. Then place burgers on grill and let sear for about 2 minutes. Flip burgers, shut lid on grill, and reduce heat to medium. This will continue to cook burgers more evenly. Remove burgers when level of doneness has been reached using an instant-read thermometer, about 145°-160°. Remove from grill and let rest, 8-10 minutes.