Whisk buttermilk and egg in a bowl. Add chicken, coat well, cover, and chill for at least 30 minutes or overnight.
Set oven to 425°F (220°C). Line a baking sheet with foil and place a greased wire rack on top.
Mix flour, salt, pepper, paprika, garlic powder, and cayenne in a shallow bowl. Dredge each chicken piece until fully coated.
Place chicken on the rack. Lightly spray tops with cooking oil.
Bake for 40–45 minutes, flipping halfway, until crispy and cooked through (165°F internal temp).
Heat oil in a saucepan. Stir in cayenne, brown sugar, paprika, garlic powder, and salt. Cook 1–2 minutes, then remove from heat.
Brush hot oil over cooked chicken. Serve with pickles and soft white bread.