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Melt enough shortening (over low heat) to come just 1/3-inch up the side of a 12-inch cast iron skillet or heavy fry pan. Once shortening liquefies raise heat to 325 degrees F. Do not allow oil to go over 325 degrees F.
Place chicken skin side down into the pan. Put thighs in the center, and breast and legs around the edge of the pan. The oil should come half way up the pan. Cook chicken until golden brown on each side, approximately 10 to 12 minutes per side. More importantly, the internal temperature should be right around 180 degrees. (Be careful to monitor shortening temperature every few minutes.)
Drain chicken on a rack over a sheet pan. Don't drain by setting chicken directly on paper towels or brown paper bags. If you need to hold the chicken before serving, cover loosely with foil but avoid holding in a warm oven, especially if it's a gas oven.