Slice the chicken breasts in half by slicing them horizontally to make two thinner cutlets. Place them between two sheets of plastic wrap and pound them to an even thickness using the flat side of a meat mallet or rolling pin (about 1/2 inch). Pat them dry with paper towels and season both sides with ½ a teaspoon each of salt and black pepper.
Whisk the flour and 1/2 teaspoon each of salt and pepper in a large shallow bowl. In a separate large shallow bowl, whisk the eggs with 1/4 cup of water. Lastly, stir the breadcrumbs, 1/2 cup of parmesan cheese, and 1/2 teaspoon each of salt and pepper in a separate large shallow bowl.
Preheat the vegetable or canola oil in a large skillet over medium heat until it registers 350°F, as measured with a digital thermometer. Place a cooling rack over a large-rimmed baking dish. Start by dredging the chicken in flour, making sure to coat the entire surface. Then, dip it in the egg, allowing the excess to drip off. Transfer to the breadcrumb mixture, gently pressing the breadcrumbs onto the chicken to ensure they stick.
Place each piece of breaded chicken on the prepared cooling rack and repeat with the remaining chicken cutlets.
Once the oil has come to temperature, cook the chicken, 2-3 pieces at a time, for 4-5 minutes per side (8-10 minutes total) or until golden and crispy. Transfer the chicken to a paper towel-lined plate to drain the excess grease, and repeat with the remaining chicken. See notes.
Preheat the oven to 350°F and spray a 9×13-inch casserole dish with non-stick cooking spray OR line a large baking sheet with foil.
Transfer the chicken to the prepared baking dish (if they overlap slightly, that's okay). Spoon approximately ¼ cup of sauce over each piece of chicken, then sprinkle with parmesan and mozzarella cheese.
Bake the chicken (if necessary) in the preheated oven until the chicken registers 160 degrees Fahrenheit as measured by a digital meat thermometer. Then low broil for 3-5 minutes or until the cheese is bubbly and golden. Serve with additional sauce on the side and fresh basil and crushed red pepper flakes for garnish if desired.