Slice each chicken breast in half horizontally so you have 4 thinner cutlets. Place each cutlet between two sheets of plastic wrap or in a zip-top bag and pound to an even 1/2-inch thickness using a meat mallet or rolling pin. This step matters because uneven chicken cooks unevenly. Season both sides lightly with salt and pepper.

Spread the flour in a shallow bowl or on a plate. Dredge each chicken cutlet in the flour, pressing lightly so it coats the surface, then shake off any excess. You want a thin, even coating, not a thick crust. Set the floured cutlets on a clean plate.

Heat 3 tablespoons of olive oil in a 12-inch skillet over medium-high heat until the oil shimmers. Add the chicken cutlets in a single layer, working in batches if the pan is crowded. Cook for 3 to 4 minutes per side until golden brown and cooked through. The internal temperature should hit 165 degrees F. Remove the chicken to a plate and cover loosely with foil.

Reduce the heat to medium. Add 1 tablespoon of the butter to the same pan and let it melt. Add the minced garlic and cook for about 30 seconds, stirring, until it smells good but has not browned. Pour in the white wine and chicken broth, scraping up any browned bits on the bottom of the pan with a wooden spoon. Let the sauce simmer for 3 to 4 minutes until it reduces by about one third.

Stir in the lemon juice and capers. Taste the sauce and adjust salt if needed. Add the remaining 2 tablespoons of butter and swirl the pan or stir until it melts into the sauce and makes it look glossy and slightly thickened.

Return the chicken cutlets to the pan and spoon the sauce over them. Let everything warm together for about 1 minute. Serve immediately with chopped parsley scattered on top and lemon slices on the side. Angel hair pasta, egg noodles, or roasted asparagus all work well underneath.