Heat oven to 350°F. Bake biscuits as directed on can.
Meanwhile, melt 1 tablespoon of the butter. In food processor, place chicken, Cajun seasoning, red pepper, salt, black pepper, 1 tablespoon of the flour, 1 tablespoon of the dry dressing mix and melted butter. Cover; process with 10 to 15 on-and-off pulses until chopped. Shape mixture into 8 patties, 2 1/2 inches in diameter.
Heat 12-inch nonstick skillet over medium heat. Melt 1 tablespoon of the butter in skillet. Add patties; cook about 5 minutes on each side or until light golden brown. Remove from skillet; cover with foil.
Stir 1/2 cup of the chicken stock into skillet, stirring to loosen bits from bottom of skillet. In small bowl, beat remaining 1/2 cup chicken stock, remaining 1 tablespoon flour and remaining 1 tablespoon dry dressing mix with wire whisk until blended; beat into mixture in skillet. Heat to boiling. Stir in hot sauce; return patties to skillet. Reduce heat to medium-low. Cover; simmer 6 to 10 minutes, turning patties once, until thermometer inserted in center of patties reads 165°F. Meanwhile, using vegetable peeler, peel carrots into thin strips.
Remove patties from skillet. Remove skillet from heat; stir remaining 1 tablespoon butter into sauce. Split biscuits; fill with patties, sauce, carrots and cheese.