Cut each chicken breast in half lengthwise. Place each half of a chicken breast between 2 sheets of plastic wrap; pound with the smooth side of a meat mallet to ¼-inch thickness.
Place flour in a shallow bowl. Season the flour with about ¾ teaspoon of kosher salt and ¼ teaspoon of ground black pepper; stir to combine.
Place the eggs in a separate shallow dish.
In a third shallow dish, combine the breadcrumbs, Parmesan cheese, and parsley.
Working with one chicken breast at a time, dredge chicken in the flour, shaking off any excess. Dip in the egg, shaking off any excess again. Finally, dredge in the breadcrumb mixture, coating evenly on both sides.
Place the coated chicken on a plate and refrigerate for 20-30 minutes, if time allows.
Preheat oven to 200°F.
Heat 1 tablespoon of oil and 1 tablespoon of butter in a large skillet over medium heat. Working in batches so that you don’t overcrowd the pan, add the chicken to the skillet and cook until crisp and golden brown, about 4-5 minutes per side.
Transfer the chicken to a baking sheet. Season with additional salt and pepper, if desired. Keep warm in the oven while you cook the remaining chicken.
Repeat the cooking process in the skillet with the remaining oil, butter, and chicken breasts.
Garnish with additional chopped fresh parsley and lemon wedges; serve.