Preheat the grill to medium-high heat. If using a grill pan, heat it over medium-high on the stove.
In a mixing bowl, combine the Cajun seasoning, olive oil, salt, and pepper. Rub this mixture all over the chicken breasts until well coated.
Once the grill is hot, place the chicken breasts on it. Grill for about 6-7 minutes on each side, or until the internal temperature reaches 165°F (75°C).
While the chicken is grilling, prepare the Alabama White Sauce. In a separate bowl, whisk together the mayonnaise, apple cider vinegar, lemon juice, Dijon mustard, garlic powder, black pepper, and cayenne pepper until smooth.
Once the chicken is cooked, remove it from the grill and let it rest for 5 minutes. This allows the juices to redistribute within the meat.
Slice the chicken and serve it drizzled with Alabama White Sauce on top or on the side for dipping.