To make 3 even chicken cutlets from each boneless skinless chicken breast: first use a sharp knife to cut the thickest part of the breast crosswise, separating it from the thinner, tapered end near the halfway point. Then carefully slice the thick part of the breast in half horizontally.
Place cutlets between layers of plastic wrap or inside a plastic, zip-top bag and pound to an even 1/4 to 1/2-inch thickness with a meat mallet or rolling pin. Season both sides of each cutlet with salt and pepper.
Place flour in a shallow bowl. Dredge one chicken cutlet in flour, shaking off any excess flour and transfer to clean plate. Repeat with remaining cutlets. (Discard remaining flour.)
Heat 2 tablespoons of the olive oil and 1 tablespoon of the butter in a 12-inch nonstick skillet over medium-high heat until hot. Add 6 chicken cutlets in a single layer; cook until crisp and golden brown, 3-4 minutes. Gently flip and cook 3-4 minutes more, or until golden brown and just cooked through. Transfer to serving platter and tent with foil. Repeat with remaining chicken cutlets.
Add chicken broth, wine, lemon juice, and capers to same skillet and bring to boil. Cook and stir, scraping up any browned bits, until sauce has thickened, about 3 minutes. Remove from heat, add remaining 2 tablespoons butter and stir to melt; spoon sauce over chicken cutlets. Garnish with parsley and lemon slices if using and serve.