Slice each chicken breast in half horizontally to create four thin cutlets. Season with salt and pepper. Dredge each piece lightly in flour, shaking off the excess.
Heat olive oil and 1 tablespoon butter in a large skillet over medium-high heat. Add the chicken and cook for 3–4 minutes per side, until golden and cooked through. Transfer to a plate and keep warm.
In the same skillet, add another tablespoon of butter and the minced garlic. Sauté for 30 seconds until fragrant. Pour in the white wine (or chicken broth) and lemon juice, scraping up any browned bits from the bottom of the pan.
Add the capers and simmer the sauce for 2–3 minutes to reduce slightly. Stir in the final tablespoon of butter for a silky finish.
Return the chicken to the skillet and spoon the sauce over the cutlets. Simmer for 1–2 minutes to reheat and blend the flavors.
Serve the chicken topped with sauce, garnished with fresh parsley and lemon slices if desired.