Place the chicken breasts between sheets of baking paper and gently pound them into four even cutlets.
Whisk together the buttermilk, egg, and hot sauce. Add the chicken and marinate for at least 20 minutes.
Combine the flour, cornstarch, Cajun seasoning, smoked paprika, garlic powder, onion powder, black pepper, and salt in a shallow dish.
Dip each chicken cutlet into the marinade, coat it in the flour mixture, dip it again, and press it into the flour a second time.
Rest the breaded chicken on a wire rack for 5 minutes while heating about 1/2 inch of vegetable oil in a heavy skillet.
Fry the chicken in batches for 4 to 5 minutes per side, or until golden and the thickest part reaches 74°C. Drain on a wire rack.
Mix the mayonnaise, grated garlic, lemon juice, Cajun seasoning, and a pinch of salt to make the aioli.
Toast the buns, spread them with garlic aioli, and layer with lettuce, crispy chicken, and pickles. Serve immediately.