If you're starting with thick chicken breasts, you will need to slice them in half horizontally (like splitting a deck of cards) to make them thinner.
Line a large cutting board with plastic wrap. Place the thin chicken breasts on the plastic and cover them with another sheet of plastic wrap. Pound the breasts with a mallet or skillet to thin them further and tenderize them.
Season the chicken breasts with salt and pepper.
In a large skillet (cast iron works great if you have one) melt one tablespoon of butter and one tablespoon of oil over high heat. Add the chicken breasts in a single layer. (They should fit, but if not, cook them in two batches.) Sauté the chicken over high heat until golden brown on the first side. (It should take 2-4 minutes.)
When the first side is done, flip over the chicken to brown the other side. (It should take 2-4 minutes.)
Turn off the heat and remove the cooked chicken to a serving platter, leaving the pan juices in your skillet. Cover the chicken with foil.
Serve the chicken topped with heated tomato sauce, if desired. Store leftovers covered in the refrigerator for up to four days.
Notes
It saves time to buy thin chicken cutlets. If you only have thick chicken breasts, you will need to slice them in half horizontally (like splitting a deck of cards) to make them thinner before starting this recipe.