Place the chicken breasts between two sheets of plastic wrap or wax paper. Pound them to an even thickness, about 1/2 inch thick, using a meat mallet or the bottom of a heavy skillet.
Season the chicken breasts with salt and pepper on both sides.
Set Up Breading Station:
Set up three shallow dishes. Place the flour in one dish, beat the eggs in another dish, and combine the breadcrumbs and grated Parmesan cheese in the third dish.
Bread the Chicken:
Dredge each chicken breast in the flour, shaking off any excess.
Dip the floured chicken breasts into the beaten eggs, allowing any excess to drip off.
Coat the chicken breasts evenly with the breadcrumb mixture, pressing gently to adhere.
Fry the Chicken:
Heat the vegetable oil in a large skillet over medium-high heat. Once hot, add the breaded chicken breasts to the skillet, making sure not to overcrowd the pan. You may need to cook the chicken in batches.
Fry the chicken for 3-4 minutes per side, or until golden brown and cooked through. The internal temperature should reach 165°F (75°C). Transfer the cooked chicken to a paper towel-lined plate to drain excess oil.
