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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.
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Meanwhile, whisk flour and cornmeal in a shallow dish. Whisk the remaining 1 tablespoon hot sauce and 1 tablespoon vinegar in a small bowl.
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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.
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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.
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Transfer to a serving platter. Repeat with the remaining 1 tablespoon oil and chicken, reducing the heat if necessary. Transfer to the platter.
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Drizzle the chicken with the reserved hot sauce mixture. Serve with carrots and celery.