If your chicken breasts are thick, place them between two sheets of plastic wrap or in a zip-lock bag and pound them to an even half-inch thickness using a meat mallet or a rolling pin. Even thickness is the key to even cooking. Pat the cutlets dry with paper towels, then season both sides with salt and pepper. Spread the flour on a plate and dredge each cutlet lightly on both sides, shaking off any excess. A thin flour coat gives you a better golden crust and helps the sauce cling later.

Heat a large skillet over medium-high heat until it is hot, about 2 minutes. Add the olive oil and 1 tablespoon of the butter. When the butter stops foaming and the oil is shimmering, lay the chicken cutlets in the pan without crowding them. Work in batches if needed. Cook for 4 to 5 minutes on the first side without moving them until a deep golden crust forms on the bottom.

Flip each cutlet with tongs and cook for another 3 to 4 minutes on the second side until golden and cooked through. The chicken is done when it feels firm to a light press and the internal temperature reaches 165 degrees F on an instant-read thermometer. Transfer the cooked cutlets to a plate and let them rest while you build the sauce. Do not wipe out the pan.

Reduce the heat to medium. Add the sliced garlic to the drippings in the pan and cook for about 30 seconds until it just turns golden, stirring constantly so it does not burn. Pour in the chicken broth and use a wooden spoon to scrape up all the browned bits from the bottom of the skillet. Add the lemon juice and capers and let the sauce simmer for 2 to 3 minutes until it reduces by about a third and smells bright and tangy.

Remove the pan from the heat and swirl in the remaining 2 tablespoons of cold butter, one tablespoon at a time, until the sauce looks glossy and slightly thickened. Taste and adjust salt and lemon as needed. Return the resting chicken to the pan or arrange the cutlets on a serving platter and spoon the sauce generously over the top. Scatter with fresh parsley and add lemon slices on the side. Serve immediately over egg noodles, rice, or alongside roasted potatoes.