Pound and season the chicken. Pat the chicken cutlets dry with paper towels. Place them between two large sheets of plastic wrap and pound with a meat mallet or the bottom of a heavy pan until thin and evenly thick, about 1/4 inch. In a small bowl, combine the salt, garlic powder, onion powder, paprika, and pepper. Remove the plastic wrap and season both sides of the chicken evenly.
Set up the breading station. Place the flour in a shallow bowl. In a second shallow bowl, whisk the eggs and Dijon mustard until smooth. In a third shallow bowl, combine the panko, Italian breadcrumbs, Italian seasoning, and Parmesan.
Bread the chicken. Working with one cutlet at a time, dredge it in flour and gently tap off the excess. Dip into the egg mixture, letting the excess drip off, then press firmly into the breadcrumb mixture to coat both sides well. Transfer to a wire rack set over a baking sheet. Let rest for 10–15 minutes before frying to help the coating adhere.
Fry the chicken. Add both oils to a large skillet and heat over medium heat. Heat to about 350°F. The oil should shimmer, and a breadcrumb dropped in should sizzle immediately without burning. Working in batches, fry 1–2 cutlets at a time for 1–2 minutes per side, until deeply golden and cooked through (160°F with a food thermometer). Flip only once.
Drain and rest. Transfer the chicken to a clean wire rack and immediately season with a little more salt. Let rest for 5 minutes, then thinly slice if desired.
Serve. Serve immediately with lemon wedges for squeezing over the top and the quick arugula salad from Note 1, if desired.
Notes
Note 1: Quick Arugula Salad: Toss 2 ounces arugula with 1 cup halved cherry tomatoes and shaved Parmesan to taste. For the dressing, shake 1 tablespoon white balsamic vinegar, 1/2 tablespoon white wine vinegar, 1/2 tablespoon Dijon mustard, 1/2 tablespoon honey, 2 tablespoons olive oil, and 1/8 teaspoon salt in a mason jar until emulsified.Storage: Chicken Milanese is best right out of the oil and doesn’t store particularly well. You can, however, bread the chicken up to 4 hours in advance and refrigerate until ready to fry.