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.
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