Prepare the Chicken: If the chicken breasts are thick, slice them lengthwise to create thinner pieces. Season both sides of the chicken with salt and pepper. Dredge each piece in flour, ensuring it is lightly coated, and shake off any excess.
Cook the Chicken: In a large skillet, melt 2 tablespoons of butter along with the olive oil over medium-high heat. Add the chicken pieces to the skillet and cook until golden brown, about 3 to 4 minutes on each side. Once cooked, remove the chicken from the skillet and set aside.
Make the Sauce: Remove the skillet from the heat and pour in the lemon juice, dry white wine (or chicken broth), and capers. Use a spatula to scrape up any browned bits from the bottom of the skillet to enhance the flavor. Return the skillet to the heat and bring the mixture to a boil. Taste the sauce and adjust seasoning with additional salt and pepper, if desired. Place the chicken back into the skillet and simmer for approximately 5 minutes.
Finish the Sauce: Add the remaining 2 tablespoons of butter to the skillet and whisk continuously for about a minute, allowing the sauce to thicken slightly.
Serve: Drizzle the sauce over the chicken and garnish with freshly chopped parsley and lemon wedges. Serve immediately.