Step 1: Place each chicken breast between two sheets of plastic wrap or parchment paper. Gently pound them to an even thickness of about 1/2 inch using a meat mallet or rolling pin. Season both sides of the chicken breasts with salt and pepper.
Step 2: Place the flour in a shallow dish. Dredge each chicken breast in the flour, shaking off any excess.
Step 3: In a large skillet over medium-high heat, heat 2 tablespoons of olive oil and 2 tablespoons of butter until hot and bubbly. Add the chicken breasts to the skillet in a single layer, being careful not to overcrowd the pan. Cook for about 3-4 minutes on each side until golden brown and cooked through. Remove the chicken from the skillet and set aside on a plate.
Step 4: In the same skillet, add the chicken broth and lemon juice, scraping up any browned bits from the bottom of the pan. Stir in the capers and bring the sauce to a simmer. Let it cook for about 2-3 minutes to reduce slightly. Remove the skillet from heat and whisk in the remaining 2 tablespoons of butter until melted and the sauce is smooth.
Step 5: Place the cooked chicken back into the skillet, spooning the sauce over the top to coat. Sprinkle with chopped parsley and garnish with lemon slices if desired.