Marinate the chicken: whisk buttermilk, egg, 1 tsp Cajun seasoning, and hot sauce. Add chicken, cover, and refrigerate 30 minutes to 8 hours.
Make the aioli: stir together mayonnaise, grated garlic, lemon juice, and Dijon. Season with salt and pepper. Refrigerate until needed.
Mix the coating: combine flour, cornstarch, remaining Cajun seasoning, garlic powder, smoked paprika, onion powder, cayenne, black pepper, and salt in a wide bowl.
Coat the chicken: remove chicken from marinade, let excess drip off, press firmly into flour mixture on all sides to build a craggy crust. Rest on a wire rack for 5 minutes.
Fry the chicken: heat oil to 350 degrees F. Fry two pieces at a time for 5 to 6 minutes per side until golden and internal temperature reaches 165 degrees F. Drain on a wire rack.
Toast the buns cut-side down in a dry skillet for 1 to 2 minutes until golden.
Assemble: spread garlic aioli on both bun halves. Layer lettuce, fried chicken, tomato, and pickles. Top and serve immediately.