Season the chicken with salt and pepper. Place the flour in a shallow dish. Lightly coat each piece of chicken with the flour, shaking off any excess.
Heat the olive oil and butter in a large skillet over medium to medium-high heat. Add the chicken and cook until golden brown on both sides, about 3 minutes per side. Note: The chicken does not have to be cooked through at this point. It will finish cooking in the sauce.
Reduce the heat to medium-low and add the sauce to the pan. Move the chicken around so that the sauce gets under it and coats each piece.
Top each piece of chicken with ¼ of the Parmesan cheese (2 tablespoons per piece of chicken) and then ¼ of the mozzarella cheese (¼ cup per piece of chicken).
Put a lid on the pan and simmer (reduce the heat to low if needed) until the chicken is cooked through and the cheese is melted, about 10 minutes.
Sprinkle with chopped fresh parsley, if desired, and serve.