Heat the vegetable oil in a large, heavy-bottomed pan or Dutch oven over medium-high heat. Once the oil is shimmering, carefully add the chicken pieces, being careful not to overcrowd the pan. Cook until the chicken is deeply browned and crispy on the outside.
Reduce the heat to medium and continue cooking for an additional 30 to 35 minutes, turning the chicken occasionally to ensure even cooking. The chicken is done when it reaches an internal temperature of 150-155 degrees F and the juices run clear. Remove to a wire rack to drain any excess oil.