Place each chicken breast between two sheets of plastic wrap and gently pound with a mallet or rolling pin until about 1/2 inch thick; this helps them cook evenly and stay tender. Set aside.
In a shallow dish, mix the flour with a pinch of salt and freshly ground pepper. Dredge each chicken piece in the flour, shaking off any excess, creating a light, even coating.
Heat the olive oil and 1 tablespoon of butter in a large skillet over medium heat until shimmering and fragrant. Carefully lay the chicken breasts in the pan, being careful not to overcrowd.
Cook the chicken until golden brown and cooked through, about 3–4 minutes per side. Once done, transfer to a plate and tent with foil to keep warm.
In the same skillet, add the wine or broth to deglaze, scraping up the browned bits with a spatula. Let it simmer for 1–2 minutes until slightly reduced and fragrant.
Add the rinsed capers to the skillet, along with the lemon zest and juice. Stir gently and cook for another 2 minutes, allowing the sauce to thicken slightly and become glossy.
Return the chicken to the skillet, spooning the sauce over the top. Cook for another minute to reheat and coat the chicken in the flavorful sauce.
Remove from heat, garnish with additional lemon slices or parsley if desired, and serve immediately. The chicken should be juicy, tender, and coated in a bright, glossy sauce.
