Place chicken breasts between two sheets of plastic wrap and pound to an even 1/2-inch thickness using a meat mallet or heavy pan. This ensures even cooking. Pat dry thoroughly with paper towels, then season both sides with 3/4 teaspoon salt and the pepper. Let sit at room temperature while you set up the breading station.

In a shallow bowl, spread the flour. In a second bowl, beat the eggs with a pinch of salt. In a third bowl, combine panko, Parmesan, garlic powder, oregano, and remaining 1/4 teaspoon salt. Mix the breadcrumb mixture well with your fingers so the cheese and seasonings are evenly distributed.

Working with one piece at a time, dredge chicken in flour and shake off excess. Dip in beaten egg, letting excess drip back into the bowl. Press firmly into the panko mixture, turning and pressing until evenly coated. Place breaded cutlets on a wire rack or large plate. Do not stack.

Heat oil and butter in a large heavy skillet over medium-high heat until shimmering and a breadcrumb sizzles immediately when dropped in, about 350 degrees F. The butter adds flavor and helps browning, but the oil prevents burning.

Add 2 cutlets to the pan without crowding. Cook until deep golden brown on the first side, 3 to 4 minutes. Flip carefully with tongs or a thin spatula and cook until the second side is equally golden and the internal temperature reaches 165 degrees F, another 2 to 3 minutes. Transfer to a wire rack set over a baking sheet to drain. Repeat with remaining cutlets, adding more oil if the pan looks dry.

Let cutlets rest for 2 minutes before serving - this keeps the crust crisp and the juices inside. Serve with lemon wedges for squeezing over the top.