Transfer chicken to a plate lined with paper towels; pat dry; discard broth.
Pour buttermilk into a large shallow bowl.
In a sealable plastic bag, combine the flour, 1 teaspoon salt, the paprika and the pepper.
Working a few pieces at a time, dip chicken in buttermilk; shake off any excess liquid; then drop into bag with flour mixture; seal bag and turn to coat pieces.
Remove pieces from bag, shake off any excess flour and transfer to a large plate; let stand at room temperature for 15 minutes.
Meanwhile, pour oil into a large, deep cast-iron skillet fitted with a deep-frying thermometer until it comes just halfway up the sides.
Heat oil to 365°F.
Cook chicken pieces 3 or 4 at a time for 8 to 10 minutes; turning every minute or so, depending on how thick the pieces are; wings will cook faster and drumsticks will take longest.
Be sure to keep oil temperature between 350°F and 365°F; the chicken is cooked when internal temperature reaches at least 165°F.
Using tongs, lift chicken out of oil and drain on paper towels.
Let cool for at least 15 minutes (chicken should still be hot, but not scorching).
Season again with a little salt and transfer to a platter; serve hot, with the dipping sauce.