Put flour, salt, pepper, paprika and cayenne pepper in a plastic bag. Shake to mix the seasoning.
Add cut-up chicken pieces, with or without bones, to bag. (Smaller pieces cook more quickly, so consider cutting each breast half in two.) Shake to coat chicken with the seasoned flour.
Heat 3/4 inch oil in a heavy frying pan (preferably cast-iron) over medium-high heat, until it gets to 350 degrees F. Use a 2-inch-deep pan if possible to reduce spatters.
When oil is hot, shake any excess flour off the chicken and add the pieces to the pan a few at a time. Don't allow the pieces to touch in the pan. Fry the chicken in batches if your pan won't hold them all with space to spare.
Cook the chicken pieces for about 10 minutes per side. Regulate the heat as the chicken cooks, keeping the oil temperature below 375 degrees - use a deep-frying thermometer to the check the temperature.