Place oil in a large fryer, Dutch oven or another thick-walled iron cooking pot with a tight-fitting lid; heat to 350º.
Combine 1 cup flour, 2 teaspoons salt and pepper in a shallow dish, stirring well with a whisk.
Place remaining 1/3 cup flour and nuts in a food processor; process until finely ground. Add peanut mixture to flour mixture, stirring well.
Combine buttermilk and eggs in a shallow dish, stirring well.
Season chicken evenly with remaining 1 teaspoon salt; dredge in flour mixture.
Dip chicken in buttermilk mixture; dredge in flour again. Fry chicken in hot oil 15 minutes or until done; remove to a paper towel-lined plate. Serve immediately.