Gently pound chicken breasts between sheets of plastic wrap with a flat meat pounder or a rolling pin until 1/3-inch thick.
Whisk together buttermilk, hot sauce, 1/2 teaspoon salt, and black pepper in a shallow dish, then add chicken, turning to coat, and let stand 15 minutes.
Place flour in a shallow dish.Stir together bread crumbs, cayenne, and remaining 1/2 teaspoon salt in another shallow dish. Remove chicken from buttermilk 1 piece at a time, dredge in flour, dip back into the buttermilk, letting excess drip off, then dredge in bread crumbs, pressing gently to help them adhere. Transfer chicken as coated to a sheet of waxed paper.
Heat oil in a 10-inch heavy nonstick skillet over moderately high heat until hot but not smoking (about 360 degrees F), then fry chicken 2 pieces at a time until golden brown and cooked through, about 3 minutes on each side. Transfer chicken with tongs to paper towels to drain and serve with hot sauce on the side.