Place the chicken breasts between two pieces of plastic wrap. Use a meat mallet or rolling pin to pound them to an even thickness. They will spread as you pound them. If the breasts are very thick, slice them in half lengthwise before pounding.
Set up two shallow dishes: One with the beaten eggs and a pinch of salt and pepper. One with the breadcrumbs, Parmesan cheese, and a pinch of salt and pepper
Have a cutting board ready for after breading to place them on.
Preheat the oven to 250°F (120°C) or a warm setting. Place an ovenproof dish inside large enough to hold the cooked cutlets. This will keep them warm while you cook in batches.
Dip one chicken breast into the egg mixture, coating both sides. Let any excess drip off, then transfer to the breadcrumb mixture. Using your dry hand, coat both sides well. Place on the board or rack. Repeat with the remaining chicken.
Heat enough oil in a large frying pan/skillet to coat the bottom of the pan. Heat over medium-high until the oil shimmers (about 350°F / 175°C). Do not let it smoke.
Carefully add the chicken to the pan, placing it into the oil away from you to avoid splashing. Cook in batches, without overcrowding. Brown on the first side for 3–4 minutes, then flip and cook for about 2 minutes, until golden brown and cooked through.
Transfer the cooked cutlets to the warm dish in the oven and repeat with the remaining chicken.
Serve warm with fresh lemon wedges and a simple vegetable side dish.