In a large non-reactive bowl, whisk together the buttermilk, 1 tablespoon of the dried basil, and the hot pepper sauce. Add the chicken pieces, turn to coat, and marinate, covered, at room temperature, stirring once or twice, for 2 hours. In a wide, deep plate (like a pie plate), combine the flour, the remaining 2 tablespoons dried basil, the salt and the pepper.
One at a time, lift the chicken pieces from the marinade, letting most of the excess drip back into the bowl. Dredge the chicken pieces in the seasoned flour. Transfer the chicken to a rack and let stand for 30 minutes to firm the coating. In 2 large skillets over medium-high heat, melt enough shortening to come 1/2-inch up the sides.
Heat the shortening until very hot (almost smoking), then carefully add the chicken pieces skin side down. Cook, uncovered, for 12 minutes. Turn the chicken and cook another 12 to 15 minutes, until done. Drain the chicken on paper towels. (To use less paper towels, drain on a pile of newspaper covered with one layer of paper towels.) Arrange the chicken on a platter, garnish with fresh basil sprigs and serve hot, warm or cool.