Whisk cashew butter, red pepper flakes, garlic, coconut aminos and ginger together. Remove ⅓ of the mixture and set aside. Gently fold ground chicken into the remaining ⅔ of the cashew sauce.
Form chicken mixture into 4 burger patties and grill until cooked through thoroughly, about 6 to 7 minutes per side.
While the patties are cooking, use a vegetable peeler to make long, thin strips of the carrot and cucumber. Place in a small mixing bowl, add minced cilantro, lime juice, and salt, and then mix to combine.
Once the burgers are cooked through, set on a plate to rest for 5 minutes, assemble burgers by spreading some of the remaining cashew sauce on each patty, evenly dividing the cucumber carrot salad between each one, and wrapping each burger in lettuce leaves.