If you are starting with whole chicken breasts, place each one on a cutting board and slice it in half horizontally so you end up with two thinner pieces. Lay each piece between two sheets of plastic wrap and pound gently with a meat mallet or rolling pin until the cutlet is an even 1/4 inch thick. Uniform thickness means they cook evenly with no raw center and no dry edges. Pat every cutlet completely dry with paper towels, then season lightly with salt and pepper on both sides.

Set up your dredging station with three shallow bowls in a row. In the first bowl, add the flour. In the second bowl, whisk together the eggs and milk until smooth. In the third bowl, stir together the breadcrumbs, grated Parmesan, garlic powder, onion powder, smoked paprika, salt, and pepper until evenly combined. Mix the Parmesan into the breadcrumbs well so it distributes through every bite.

Working one cutlet at a time, dredge it in the flour and shake off any excess. You want just a thin, even layer of flour, not a thick coating. Next, dip it into the egg wash and let any drips fall back into the bowl. Finally, press it firmly into the Parmesan breadcrumb mixture, turning it over and pressing again so the coating adheres to every part of the surface. The pressing step is important because a crust that is just patted on will slide off in the pan. Place each coated cutlet on a wire rack and let them rest uncovered for 5 minutes. This short rest lets the coating bond to the chicken.

Set a large heavy skillet over medium to medium-high heat and add the oil. Let the oil heat until it shimmers and a pinch of breadcrumbs dropped in sizzles immediately, about 2 minutes. Add the butter and let it melt into the oil. Work in batches rather than crowding the pan. Crowding drops the oil temperature and leads to steaming instead of frying, which gives you a soggy crust. Add 2 cutlets and cook for 3 to 4 minutes on the first side without moving them. You want deep golden brown color before you flip, so resist the urge to peek too early.

Flip each cutlet once and cook the second side for another 3 to 4 minutes. The crust should be evenly golden and feel firm when you press it lightly. If you have an instant read thermometer, the center should read 165 degrees F. Transfer the finished cutlets back to the wire rack, not to a plate, so air can circulate underneath and the crust stays crunchy. Add a splash of fresh oil to the skillet between batches and let it come back up to temperature before adding the next round. Serve immediately with lemon wedges on the side.