Preheat the oven to broil. Place a wire rack inside each of 2 rimmed baking sheets.
Cut the chicken breasts in half horizontally through the middle (like slicing a bagel) making 4 roughly equal pieces. Place a sheet of plastic wrap on a cutting board. Put 1 chicken piece in the center of the plastic wrap and top with another sheet of plastic. Pound the chicken with the flat side of a meat mallet, starting in the center and working your way to the edges, until the chicken is 1/4-inch thick. Transfer to a prepared baking sheet. Repeat with the remaining chicken pieces.
To set up a standard breading station, mix the flour, 1 tablespoon salt and 1/2 teaspoon pepper in a shallow dish. Beat the eggs and 1/2 teaspoon salt in a second shallow dish. Place the breadcrumbs in a third shallow dish.
Dredge 1 piece of chicken in the flour mixture, turning to coat evenly. Shake off any excess, then dip into the egg mixture, turning to fully coat it on both sides. Let the excess egg drip off, then press the chicken firmly into the breadcrumbs, turning to coat on both sides and packing the crumbs into any crevices. Return the chicken to the baking sheet and repeat with the remaining chicken pieces.
Heat the oil in a large skillet over medium-high heat (you will know the oil is ready if you drop a few breadcrumbs into the skillet and they immediately sizzle). Add 2 pieces of chicken and cook until the breading is golden-brown and crisp and the chicken is just cooked through, 3 to 5 minutes per side, turning with tongs. Transfer to the other prepared baking sheet. Make sure the oil is hot again before repeating with the remaining 2 pieces of chicken.
Spread 1/4 cup vodka sauce onto each piece of chicken then top each with 1 tablespoon Parmesan and 1/4 cup mozzarella. Broil until the cheese melts and browns in spots, 3 to 5 minutes. Pour the remaining vodka sauce onto a serving platter and top with the chicken pieces. Sprinkle with the parsley.
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