Slice each chicken breast in half horizontally to create 4 thin cutlets of even thickness, about 1/2 inch thick. Place them in a bowl with the buttermilk and hot sauce, toss to coat, and let them marinate in the refrigerator for at least 30 minutes or up to 4 hours.
In a shallow dish, whisk together the all-purpose flour, cornstarch, smoked paprika, garlic powder, onion powder, cayenne pepper, salt, and black pepper.
Remove one chicken cutlet from the buttermilk, letting the excess drip off. Press it firmly into the flour mixture, coating both sides well. Dip it back into the buttermilk briefly, then press it into the flour mixture a second time for an extra-thick crust. Repeat with the remaining cutlets.
Preheat the air fryer to 380 degrees F. Spray the air fryer basket generously with cooking spray. Place the breaded chicken cutlets in the basket in a single layer without overlapping, then spray the tops of the chicken liberally with cooking spray.
Air fry for 12 to 14 minutes, flipping halfway through and giving another light spray of cooking spray on the flipped side. The chicken is done when it is deeply golden, crispy, and has reached an internal temperature of 165 degrees F.
While the chicken cooks, stir together the mayonnaise and dill pickle juice to make a tangy sandwich sauce. Toast the brioche buns lightly.
Spread the pickle mayo on both halves of each toasted bun. Layer on the butter lettuce, a crispy chicken cutlet, and dill pickle slices. Press gently and serve immediately while the chicken is hot and crunchy.