Slice each chicken breast lengthwise to yield six thin cutlets. Season both sides of the cutlets with salt and black pepper.

Place the flour in a shallow dish and coat each chicken piece lightly, shaking off any excess. Arrange the coated cutlets in a single layer on a plate.

Heat a large skillet over medium-high heat. Add 2 tablespoons of olive oil and 2 tablespoons of butter. When the butter begins to sizzle, place three chicken cutlets in the skillet. Sauté each side for 2–4 minutes, or until the internal temperature reaches 165°F. Transfer the cooked chicken to a serving platter. Repeat the process with the remaining chicken, adding 1 tablespoon of oil and 1 tablespoon of butter to the skillet as needed.

Keeping the skillet at medium-high heat, pour in the white wine, followed by the lemon juice, 1 tablespoon of butter, and the rinsed capers. Allow the mixture to simmer for 1 minute, scraping up any browned bits from the pan to infuse the sauce with additional flavor.

Dip the cooked chicken cutlets into the sauce one by one, ensuring they are well-coated, and return them to the platter. Continue simmering the sauce for an additional 2 minutes to slightly thicken.

Pour the sauce over the chicken, then garnish with lemon slices and fresh parsley before serving.