Make Chicken Cutlets - Thoroughly dry the chicken with a paper towel. Cut each piece horizontally lengthwise to make two thinner cutlets, four total.
Flatten the Pieces - Place one of the pieces between two plastic sheets or inside a large plastic bag and flatten to an even ¼-inch thickness using a meat tenderizer. Repeat with the remaining pieces. Generously season both sides with salt and pepper.
Bread the Chicken - In one shallow bowl, mix the flour, 1 teaspoon kosher salt, and ½ teaspoon black pepper. In a second shallow bowl, whisk together eggs and olive oil. In a third shallow bowl, combine panko breadcrumbs, parmesan cheese, Italian seasoning, and lemon zest.Dip the chicken in the flour, shaking off any excess coating. Next, dip in the egg mixture, coat completely, and drain any excess. Lastly, dip in the breadcrumbs, pressing to coat evenly. Repeat the process with the remaining pieces.
Pan-fry the Chicken - Heat ⅓ cup of vegetable oil in a 12-inch nonstick skillet over medium-high heat. Once shimmering, place two chicken pieces in the pan. Fry until the first side is golden brown and crispy, 2 to 3 minutes. Turn the heat down to medium and flip the chicken. Cook until the internal temperature is 160 to 165ºF (71 to 74ºC) on a meat thermometer, about 2 to 4 minutes.Transfer to a paper towel-lined plate for 15 seconds to drain excess oil. Place on a wire rack set on top of a sheet pan. Discard the oil in the pan and wipe it clean. Add ⅓ cup of fresh vegetable oil, and repeat the cooking procedure with the two remaining pieces.
To Serve - Serve garnished with parsley and lemon wedges.
Reheating: Reheat individual pieces in the microwave on high power in 15 to 30-second intervals until hot. Alternatively, place on a foil-lined sheet pan and bake at 375ºF (191ºC) until hot and crispy, about 10 to 15 minutes.
Using Smaller Pieces: Four chicken breasts, about 6 to 8 ounces, can be used. Flatten until ¼-inch thick.
Bread crumbs: Traditional fine Italian breadcrumbs can be used but won’t be as crunchy.
Make Ahead: The chicken can be breaded, covered, and refrigerated for 1 day before cooking.
Baking: Bake chicken in the middle of the oven at 450ºF (232ºC) on top of a wire rack set on a foil-lined sheet pan until the internal temperature reaches between 160 to 165ºF (71 to 74ºC), about 15 to 20 minutes.
Make it Gluten-Free: Use gluten-free flour and breadcrumbs.