To make deep-fried chicken, prepare 2 chicken drumsticks and 2 chicken thighs and leave the skin on them. Coat the meat in egg and a seasoned flour mixture and fry in vegetable shortening.
Place the chicken in a medium-sized pan, and add 3 to 4 cups milk. Seal in plastic wrap, and refrigerate for 2 hours. Set a large saucepan on a medium-high heat. With tongs, place the chicken into the saucepan, followed by the milk. Boil and then lower the heat to medium-low. Allow the chicken to cook for approximately 20 minutes. Separate the chicken from the milk, and set on a rack to cool for approximately 15 minutes.
Scoop 2 1/4 cups vegetable shortening into a Dutch oven, and heat at 325 degrees Fahrenheit. Shake 1 tablespoon and 1 teaspoon salt along with 3/4 cup all-purpose flour and 1 teaspoon cayenne pepper in a large plastic bag. Beat 1 egg in a medium-sized bowl. Individually put each piece of chicken into the plastic bag, and shake. Remove the chicken, and coat with the egg. Return the chicken to the bag, and toss again.
Put the coated chicken into the heated shortening. Fry until crisp and brown in color for approximately 1 minute on each side. Let the chicken drain on a plate covered with a paper towel and serve.