Rinse chicken well, and pat pieces dry. Slowly heat the shortening or oil in a Dutch oven (preferable) or heavy cast-iron skillet over moderate heat while preparing the chicken.

Whisk together the flours, 1 tablespoon cayenne pepper, 1 tablespoon Old Bay seasoning, 1 tablespoon dried rosemary, and 1 tablespoon each salt and pepper to combine. Place seasoned flour in a brown paper bag. Whisk together the eggs and milk in a bowl and set aside. Season the chicken pieces with the remaining cayenne, Old Bay seasoning, rosemary, salt and pepper. Dip the chicken pieces in egg mixture, let excess drain off, then place pieces in bag of seasoned flour (about 4 at a time). Remove pieces from bag, dip in egg mixture once more, then return to seasoned flour. Shake to coat well. Place coated chicken on a wire rack for a few minutes to set coating.

Add about one-third of the chicken pieces to the hot (360-degree) oil and, after 1 minute, turn pieces over. Continue to cook, uncovered (8 to 10 minutes longer for white meat, 13 to 15 minutes longer for dark meat), turning pieces every 5 minutes or until well-browned and juices run clear.