Cut chicken into eight pieces (two each of breasts, thighs, drumsticks, and wings). Set aside.
In large bowl, whisk together 4 cups buttermilk, 1/2 cup salt, sugar, garlic, 2 tablespoons celery seed, paprika, and 2 teaspoons cayenne pepper. Add chicken and turn to make sure each piece is coated. Cover and refrigerate at least 1 hour and up to 4.
After chicken has marinated, rinse pieces and pat dry lightly. Discard buttermilk marinade.
In a large bowl, whisk together flour, remaining tablespoon salt, remaining tablespoon celery seed, remaining teaspoon cayenne, and ground pepper. Place remaining 2 cups buttermilk in a separate bowl.
Heat oil (about 2 quarts) in a large Dutch oven to 325 F. Working one piece at a time, add chicken to flour and shake to coat. Remove and dip in buttermilk, then back in the flour. Shake off excess and carefully place in oil.
Fry until golden brown and chicken is cooked through; this will take 5 to 10 minutes, depending on size of the piece. Repeat with remaining chicken.