Sprinkle the chicken with salt and pepper and set aside. Place the flour in a shallow plate and season with the cayenne, salt and pepper. Set aside.
Pour the canola oil into a large cast-iron skillet to a depth of 3/4 inch. Heat the oil over medium-high heat until a deep-fry thermometer inserted in the oil registers 375 degrees F. Set up brown paper bags or several layered paper towels nearby for draining the chicken.
Starting with the dark meat, one piece at a time, add the chicken to the seasoned flour and turn to coat. Shake to remove excess flour. One piece at a time, add the chicken to the hot oil. The oil should not quite cover the chicken. (Do not overcrowd the pan; this can be dangerous and result in soggy chicken, too.)
Fry the pieces, turning them once or twice, until the coating is rich golden brown on all sides, 10 to 14 minutes. Reduce the heat to medium and cover the skillet. Continue cooking until the chicken is cooked all the way through and the juices run clear when pricked with a knife, another 10 to 15 minutes. (An instant-read thermometer inserted into a thigh should register 170 degrees F.) Remove and drain the chicken on the brown paper bags or paper towels.
Remove the skillet from the heat. Carefully pour off most of the grease into a metal bowl or can, leaving 2 to 3 tablespoons in the pan with the browned crumbs. Reduce the heat to very low. Add 4 tablespoons of the remaining flour used to coat the chicken. Stir together with a wooden spoon and cook, stirring often, until golden brown, about 3 minutes. Add the milk and stock or water. Cook, stirring often, until the gravy is smooth and thick enough to coat the back of a spoon. Add a heaping pinch of black pepper. Taste and adjust for seasoning with salt.