In a small bowl, mix the mayonnaise, breadcrumbs, lime juice, and hot sauce. Set aside.
In a large bowl, gently stir together the chicken, salt and pepper, chiles, jalapeño, red pepper, cilantro and red onion, grated Cheddar cheese, and garlic. Add the reserved mayonnaise mixture and stir together until just mixed. Do not overmix.
Form 6 (4-5 ounce each) patties. Place on platter, cover, and refrigerate for an hour or up to overnight for burgers to absorb liquids and firm up. If still soft, consider freezing 5-10 minutes.
Heat heavy skillet, griddle, gas or charcoal grill to medium. Brush chilled patties well with olive oil, adding a tablespoon or two to the pan if cooking on the stove. Cook 4-5 minutes. Carefully turn (this is a tender burger), add cheeses, and cook another 4-5 minutes or until instant-read thermometer reads 165 degrees Fahrenheit. Do not overcook. Serve hot on grilled buns with your choice of mayonnaise, avocado or guacamole, shredded cabbage, and sliced tomatoes. Tasty sides: grilled corn or coleslaw or Air Fryer French Fries.