Slice each chicken breast in half horizontally to create two thinner pieces. Pound the chicken pieces to an even thickness if necessary.
Season both sides of the chicken with salt and pepper.
Dredge the chicken in flour, shaking off any excess.
Cook the Chicken:
In a large skillet, heat the olive oil and 2 tablespoons of butter over medium-high heat.
Add the chicken pieces to the skillet and cook for about 3-4 minutes per side, or until golden brown and cooked through. Transfer the chicken to a plate and set aside.
Make the Sauce:
In the same skillet, add the white wine (or chicken broth) and cook for about 1-2 minutes, scraping up any browned bits from the bottom of the pan.
Add the lemon juice, chicken broth, and capers. Bring the mixture to a simmer and cook for about 3-4 minutes, allowing the sauce to reduce slightly.
Finish the Sauce:
Reduce the heat to low and stir in the remaining 2 tablespoons of butter until melted and the sauce is smooth.
Return the chicken to the skillet, spooning the sauce over the top to coat the chicken. Cook for an additional 1-2 minutes to heat the chicken through.
Serve:
Transfer the chicken to serving plates and spoon the sauce over the top.
Garnish with chopped fresh parsley and lemon slices, if desired.