Halve each chicken breast horizontally to make 4 cutlets. Lay them between two sheets of plastic wrap and pound to an even 1/2 inch thick. Even thickness is what keeps the chicken from drying out on the thin end before the thick end cooks through.

Season the cutlets on both sides with the salt and pepper. Spread the flour on a plate and dredge each cutlet, pressing lightly, then shake off the excess. You want a thin coat, not a thick crust.

Heat the olive oil and 2 tablespoons butter in a large skillet over medium-high until the butter stops foaming. Add the cutlets (work in two batches if needed so you do not crowd the pan) and sear about 3 minutes per side, until golden and just cooked through. Move them to a plate.

Pour in the white wine and scrape up all the browned bits from the bottom of the pan with a wooden spoon. Let it bubble and reduce by about half, 1 to 2 minutes.

Add the chicken broth, lemon juice, and capers. Simmer 3 to 4 minutes to concentrate the flavor slightly.

THE ONE TRICK: Pull the pan completely off the heat. Add the cold butter a few pieces at a time, swirling the pan constantly, letting each addition melt before adding more. The sauce will turn glossy, thick, and silky as it emulsifies. Do not let it boil once the butter goes in or it will break.

Stir in the parsley. Return the cutlets to the pan, spoon the sauce over them, and let them warm through for a minute. Garnish with lemon slices and serve right away.