Dry chicken breasts with paper towels. Lay them shiny side down on a cutting board. Cut off chicken tenders and set them aside. Using a sharp knife, score chicken breasts top to bottom and side to side about every half inch, and deep enough to not cut all the way through.
Cover the breasts with a piece of Saran wrap and slightly pound with a meat tenderizer to make them uniform thickness.
Season the breasts with salt and pepper on each side. In a small bowl, combine the sour cream with the mustard, then rub this mixture all over the chicken breasts, making sure it gets into every crevice. Place the breasts in a bowl, cover and refrigerate for about 15 minutes.
Whisk the egg in a medium bowl and set aside. Spread the flour on a large plate and set aside.
Heat a skillet filled with cooking oil so that it just covers the bottom. One by one, dip the chicken breasts in the egg mixture, then the flour on both sides, shake off excess, and place on the heated skillet. You will want to do the frying in batches. The first batch will be the first three chicken breasts, while the second batch will be the fourth chicken breast and the four chicken tenders.Fry chicken breasts over medium-high heat about 2-3 minutes per side, uncovered, until deep golden brown on each side.
Serve hot with your favorite side dish, or use to make some of the most delicious chicken sandwiches.