Whisk buttermilk, 2 tablespoons hot pepper sauce and 2 tablespoons vinegar in a large bowl until combined. Add chicken; toss to coat. Transfer to the refrigerator and let marinate for at least 10 minutes or up to 1 hour, stirring occasionally.
Meanwhile, whisk flour and cornmeal in a shallow dish. Whisk the remaining 1 tablespoon hot sauce and 1 tablespoon vinegar in a small bowl.
Remove the chicken from the marinade and ll in the flour mixture until evenly coated. (Discard remaining marinade and flour mixture.) Sprinkle both sides of the chicken with cayenne.
Heat 1 tablespoon oil in a large nonstick skillet over medium-high heat. Add half the chicken, placing each in a little oil. Cook until golden brown and cooked through, 3 to 4 minutes per side.
Transfer to a serving platter. Repeat with the remaining 1 tablespoon oil and chicken, reducing the heat if necessary. Transfer to the platter.
Drizzle the chicken with the reserved hot sauce mixture. Serve with carrots and celery.