In a mixing bowl, combine Cajun seasoning, paprika, garlic powder, black pepper, salt, olive oil, and lemon juice to create a marinade.
Add the chicken breasts to the marinade, ensuring they are well-coated. Cover with plastic wrap and refrigerate for at least 30 minutes, preferably up to 2 hours for better flavor.
Meanwhile, prepare the Alabama white sauce. In another mixing bowl, whisk together mayo, apple cider vinegar, Dijon mustard, lemon juice, garlic powder, black pepper, and cayenne pepper until smooth. Refrigerate until ready to serve.
Preheat the grill to medium-high heat.
Remove the chicken from the marinade and let excess marinade drip off. Place the chicken breasts on the grill.
Grill the chicken for 6-7 minutes on each side or until the internal temperature reaches 165°F (75°C).
Remove the chicken from the grill and let it rest for 5 minutes.
Serve the grilled chicken with the Alabama white sauce drizzled on top or on the side.