Place chicken breasts between parchment paper or plastic wrap. Pound to about 1/2 inch thickness. Season both sides with salt and pepper. Dredge each breast in flour, shaking off excess.
Heat 2 tablespoons olive oil in a large skillet over medium-high heat. Add two chicken breasts and cook 3-4 minutes per side until golden and cooked through. Remove and set aside. Repeat with remaining chicken and olive oil.
Reduce heat to medium. Add 2 tablespoons butter to the skillet. Once melted, pour in chicken broth and fresh lemon juice, scraping up browned bits. Stir in rinsed capers and simmer for 3-4 minutes until slightly reduced.
Turn off heat. Whisk in remaining 2 tablespoons butter until sauce is glossy and thickened. Return chicken to skillet and spoon sauce over. Sprinkle with chopped parsley and serve immediately.
