In a large bowl, soak the chicken in enough salted water to cover in the refrigerator overnight. Add about 1 tablespoon salt per cup of water. Drain the chicken and shake off the excess water. In a pie plate, mix together the flour, pepper and salt. Roll the chicken in the flour to coat. Into two 12-inch cast iron skillets, pour enough vegetable oil to come to a depth of 1 1/2 inches in each skillet. Heat over medium-high heat until very hot, 305 degrees F.
Using tongs, carefully add the chicken pieces to the hot oil, making sure there is a lot of space between the pieces. Cook all the chicken at the same time. Cover the skillets and lower the heat to medium. After 15 minutes, the bottom sides should be brown. Turn the pieces over with the tongs, and continue to cook, covered, browning on all sides, and adjusting the heat as necessary. Remove the breast pieces after 12 minutes. All the chicken should be cooked in about 30 minutes. Transfer the chicken to paper towels to drain. Serve immediately.