In a large mixing bowl, add the ground chicken, jalapeño peppers, red onions, bell peppers, cumin, oregano, salt, pepper, and egg. Using your hands, mix until the meat is well combined.
Using a measuring cup, scoop out about 1/2 cup of the meat mixture and form into a ball, then press down to shape into a patty. Repeat with the remaining meat mixture.
Heat the oil in a large heavy-bottomed skillet over medium heat. When hot, cook the patties, working in batches as needed, 4 to 5 minutes until browned. Flip and cook for 4 to 5 minutes more until the internal temperature teaches 165°F. Transfer the cooked burgers to a paper towel-lined plate and repeat with the remaining burgers.
Make the Aioli
In a food processor or small high-powered blender, add the mayonnaise, garlic, cilantro, lime zest, lime juice, salt, and pepper and blend until the mixture is just combined.
To Serve
To serve, stack one patty on a large lettuce leaf and top with sliced avocado, tomatoes, and aioli.