To make the guacamole—in a medium bowl mash the avocados. Squeeze in or add the lime juice. Season with salt. Mix in onion, cilantro, and garlic. Mix in cayenne pepper. Cover and refrigerate until needed.
In a bowl combine ground chicken, 1/3 cup cilantro, shredded cheddar cheese, garlic, jalapeno pepper, juice of ½ lime, cumin, paprika, salt, and pepper. Shape into 4 patties.
Lightly grease the grates of an outdoor grill. Heat the grill to medium-high heat. Add the chicken burgers and cook until the bottom is browned (about 3-4 minutes).
Flip the burgers and cook on the other side until browned on the bottom and fully cooked (the internal temperature has reached 165 degrees F.).
During the last few minutes of cooking, top each patty with a slice of cheddar cheese and cook until the cheese has melted. Remove from the grill.
Place burger onto the toasted buns and top with guacamole and cilantro.