Soak chicken pieces in salted water (1 tablespoon of salt per quart of water) for at least 1 hour. Pat dry.
Mix flour, paprika, salt and pepper in a large plastic bag. Coat the chicken with the seasoned flour mixture.
Heat the shortening in a large deep skillet over medium-high heat to 360 degrees F. Toss the onions in the hot oil and fry until dark brown, about 12 minutes (the browned onions flavor the oil). Shake any excess flour from the chicken pieces then add them to the onions. Fry the chicken until it is crisp and brown on the first side, about 10 minutes. Turn the pieces of chicken over and continue cooking until the juices run clear, approximately 20 minutes (remember the legs may take a little longer than the breasts). Transfer the chicken to a rack to drain. Discard the onions. Serve the chicken hot or at room temperature.