Prepare the chicken:
Slice the chicken breasts in half horizontally to create four thinner cutlets. Use a meat mallet to gently pound each piece to an even thickness of about 1/2 inch. Season both sides with salt and pepper.
Dredge the chicken:
Place the flour in a shallow bowl. Lightly dredge each chicken cutlet in the flour, shaking off any excess.
Cook the chicken:
Heat the olive oil and 1 tablespoon of butter in a large skillet over medium heat. Add the chicken cutlets and cook for 3–4 minutes per side, or until golden brown and cooked through. Transfer the chicken to a plate and keep warm.
Make the sauce:
In the same skillet, add the chicken broth, lemon juice, and capers. Whisk to combine, scraping up any browned bits from the bottom of the pan. Simmer for 2–3 minutes.
Finish the sauce:
Reduce the heat to low and whisk in the remaining 1 tablespoon of butter. Allow the sauce to thicken slightly, about 1–2 minutes.
Combine and serve:
Return the chicken to the skillet and spoon the sauce over the cutlets. Let the chicken warm through for 1–2 minutes. Garnish with fresh parsley if desired. Serve immediately.