Start by trimming any excess fat from your chicken breasts. Place each breast between two sheets of plastic wrap or inside a zip-top bag. Using a meat mallet or rolling pin, gently pound the chicken until it’s about 1/2-inch thick. Season both sides generously with salt and pepper.
Pour the all-purpose flour into a shallow dish. Lightly coat each chicken breast with flour, shaking off any excess.
Heat the olive oil and 2 tablespoons of butter in a large skillet over medium-high heat. Once the butter melts and starts to foam, add the chicken breasts. Cook for about 3-4 minutes per side, or until golden brown and cooked through. Remove the chicken from the pan and set aside on a plate.
In the same skillet, add the chicken broth, fresh lemon juice, and rinsed capers. Use a wooden spoon to scrape up any browned bits from the bottom of the pan. Let the mixture simmer for 3-5 minutes, reducing slightly.
Turn the heat to low and whisk in the remaining 2 tablespoons of butter, one tablespoon at a time, until the sauce is smooth and glossy. Return the chicken breasts to the skillet, spooning the sauce over the top. Let everything warm through for a minute or two.
Transfer the chicken to serving plates. Spoon extra sauce and capers over the chicken. Garnish with chopped fresh parsley and lemon slices. Serve immediately with your favorite sides.
