Slice each chicken breast horizontally into thin cutlets. Season both sides with salt and pepper, then dredge lightly in flour and shake off the excess.
Heat the olive oil and 1 tablespoon butter in a large skillet over medium-high heat. Cook the chicken for 3–4 minutes per side, until golden and cooked through to 165°F. Transfer to a plate.
Reduce the heat to medium. Add the garlic and cook for about 30 seconds, stirring. Pour in the wine, scraping the browned bits from the skillet, and simmer 2–3 minutes.
Add the chicken broth, lemon juice, capers, and lemon zest. Simmer for about 3 minutes, then whisk in the remaining 2 tablespoons butter until the sauce is glossy.
Return the chicken to the skillet and spoon sauce over it. Simmer 1–2 minutes to warm through. Sprinkle with parsley.
Serve the chicken and sauce over cooked pasta. Finish with extra parsley, black pepper, and Parmesan if desired.