Place 1 cup all-purpose flour, 1/3 cup corn meal, 3 tbsp corn starch, 2 tsp salt, and 1 tsp. pepper in a shallow pan or pie plate. Stir together well and set aside.
Pour buttermilk into a separate large bowl. Add the egg, and whisk together well. Set aside.
Add the vegetable oil to a large frying skillet (about 10”-12” (22-25cm) in diameter) Heat to 350° (176°).
While you are waiting for the vegetable oil to heat, roll each piece of chicken in the flour mixture, coating it well. Using tongs, transfer each piece to the buttermilk mixture, and coat well, then coat again in the flour mixture.
Place the coated piece of chicken carefully into the hot vegetable oil. Place as many pieces as possible in the skillet. You may have to fry the chicken in batches.
Allow the chicken to cook for about 15-20 minutes, then carefully flip to the other side and cook for another 15-20 minutes. Juices from the chicken should run clear and not pink.
Once chicken is well cooked, removed from the frying pan, and transfer to a pan lined with paper towels. Keep chicken warm in the oven while you prepare the cream gravy.
Turn the heat off from the frying pan, and allow to cool briefly.
Carefully pour the frying oil into a heat proof dish, leaving about ¾ cup of the frying oil in the pan, along with any of the cracklings.
Turn the heat back on a medium flame, and add 1/3 cup flour. Stir until smooth and lump free. Once the mixture is smooth, add the milk.
Continue to stir, scraping the bottom of the pan to loosen any cooked-on bits. Season with salt and pepper, stirring and simmering until thickened, about 5 minutes.