01 - In a large bowl, combine buttermilk, hot sauce, and Cajun seasoning. Add chicken breasts and ensure each piece is fully coated. Cover and refrigerate for at least 30 minutes, preferably up to 4 hours.
02 - In a separate bowl, mix together all-purpose flour, corn starch, Cajun seasoning, garlic powder, paprika, salt, and black pepper until well blended.
03 - Remove chicken breasts from the marinade and let excess liquid drip off. Thoroughly dredge each breast in the seasoned flour mixture, pressing gently to ensure an even, adhesive coating.
04 - Pour vegetable oil into a deep skillet or Dutch oven to a depth of about 2 inches. Preheat oil to 350°F. Fry chicken breasts for 6 to 8 minutes per side, or until golden brown and internal temperature reaches 165°F. Transfer to paper towels to drain excess oil.
05 - In a small bowl, whisk together mayonnaise, minced garlic, fresh lemon juice, Dijon mustard, salt, and black pepper. Refrigerate aioli until ready to use.
06 - Toast brioche buns until lightly golden. Generously spread garlic aioli on the cut sides of each bun.
07 - Layer bottom bun with lettuce, a piece of fried chicken, tomato slices, and dill pickles if desired. Crown with the top bun and serve immediately.