Fry the chicken:
In a large, shallow bowl or plate, mix the flour, salt, and pepper well with a fork. Rinse each chicken piece, pat dry, and dredge in the flour mixture to coat it well. Tap to remove excess flour. Set aside on a large plate.
In a large cast-iron skillet or other deep, heavy frying pan, heat the shortening over medium-high heat until sizzling hot but not smoking. Test the fat with a pinch of flour; if it “blooms” at once, it’s ready.
Using tongs, carefully put the floured chicken pieces in the hot fat, skin side down, without crowding the pan. Let the chicken cook until lightly browned on one side, about 5 minutes. Turn each piece and brown again, about 5 minutes. Reduce the heat slightly and continue frying, turning occasionally to brown evenly.
Arrange the wings, breasts, and legs at different angles as needed, turning the pieces about every 5 minutes to cook and brown evenly and prevent scorching. Fry until the chicken is cooked through, 20 to 25 minutes after the initial browning (for a total of 30 to 35 minutes). Check for doneness using a meat thermometer (it should read 180°F) or cut through to the bone: the juices should run clear and there should be no pink. Remove each piece as it’s done, setting it on clean brown paper grocery bags or paper towels to drain.
Make the cream gravy:
Slowly and carefully drain the grease from the cast-iron skillet into a heatproof bowl, keeping behind as many of the browned bits in the frying pan as you can. Slowly pour the grease into another heatproof bowl to capture any sray brown bits that escaped from the pan, and return the bits back to the pan. Add 2 Tbs. back to the pan as well.
Heat the pan over medium-low heat until hot. Add the flour and stir well for 1 to 2 minutes, scraping up the brown bits. Add the milk or water and increase the heat to medium. Cook, stirring occasionally, until the gravy comes to a gentle boil. Simmer, stirring often, until the gravy is smooth, shiny, and thickened. Stir in the salt and pepper, adding more seasonings to taste. Transfer to a serving bowl.
Pile the drained pieces of chicken on a platter and serve them hot, warm, or at room temperature, with or without the cream gravy.