With a cleaver or heavy knife, split the chicken breasts, then cut each breast in half again.
Place the chicken pieces in a shallow container skin side down and pour the buttermilk over them. Cover with plastic wrap and marinate in the refrigerator for at least 6 hours or overnight if possible.
Preheat the oven to 375 F.
Remove the chicken from the buttermilk and season both sides with salt and pepper.
Place the beaten eggs and flour in separate shallow bowls. Dredge the chicken pieces first in the flour, then the egg, and then the flour again.
In a 12-inch saute pan with deep sides, melt enough shortening so that it is 2 inches deep and heat the pan until the oil begins to smoke.
Fry the chicken for 2 minutes on each side, until the skin is golden brown. Transfer to a baking sheet and bake the chicken for 15 minutes, or until cooked through.
Serve warm immediately, or store in the refrigerator for the next day’s picnic.