Prepare the Chicken: Place each chicken breast between two sheets of plastic wrap. Using a meat mallet or rolling pin, pound the chicken to an even 1/4-inch thickness.

Dredge the Chicken: In a shallow dish, combine the flour, salt, and pepper. Dredge each chicken breast in the flour mixture, shaking off any excess.

Sear the Chicken: Heat 2 tablespoons of olive oil and 2 tablespoons of butter in a large skillet over medium-high heat. Add two chicken breasts to the skillet and cook for 3-4 minutes per side, or until golden brown and cooked through. Remove the chicken from the skillet and set aside. Repeat with the remaining olive oil, butter, and chicken breasts.

Deglaze the Pan: Reduce the heat to medium. Pour the white wine into the skillet and scrape up any browned bits from the bottom of the pan. Let the wine reduce slightly, about 2-3 minutes.

Make the Sauce: Add the chicken broth and lemon juice to the skillet. Bring to a simmer and cook for another 2-3 minutes, allowing the sauce to slightly thicken. Stir in the capers.

Incorporate the Cream: Reduce the heat to low. Gradually whisk in the heavy cream, ensuring it is fully incorporated into the sauce. Simmer gently for 1-2 minutes, allowing the sauce to thicken slightly. Do not boil.

Combine and Serve: Return the cooked chicken breasts to the skillet. Spoon the sauce over the chicken, ensuring it is evenly coated. Heat through for another minute or two. Garnish with fresh parsley and lemon slices (if desired). Serve immediately over pasta, rice, or alongside vegetables.