In a mixing bowl, combine the Cajun seasoning, olive oil, garlic powder, onion powder, salt, black pepper, and lemon juice.
Add the chicken breasts to the bowl, ensuring they are well-coated with the spice mixture. Allow the chicken to marinate for at least 15 minutes, or up to 2 hours for deeper flavor.
In another bowl, whisk together the mayonnaise, apple cider vinegar, lemon juice, Dijon mustard, Worcestershire sauce, black pepper, and cayenne pepper until smooth. Adjust the seasoning to your liking and set aside.
Preheat the grill to medium-high heat. Once hot, place the marinated chicken breasts on the grill.
Grill the chicken for about 6-7 minutes on each side, or until the internal temperature reaches 165°F (75°C). Avoid flipping the chicken too often to ensure good grill marks.
Once cooked, remove the chicken from the grill and let it rest for a few minutes before slicing.
Serve the grilled chicken on a plate drizzled with Alabama white sauce. Also, serve additional sauce on the side for dipping.