Cut each piece of chicken in half horizontally and place each piece between 2 sheets of plastic wrap. Using the smooth side of a meat mallet, pound to 1/4-inch thickness.
Place flour, salt and pepper in a shallow dish. Lightly coat chicken pieces with seasoned flour, tapping to remove excess.
In a large skillet, heat 2 tablespoons butter and oil over high heat, until hot but not smoking. Brown chicken in batches, turning once, about 2 minutes per side. Transfer to a serving platter and keep warm.
Remove any excess fat from the skillet and discard. Add wine, chicken broth, lemon juice, and capers. Cook over high heat until simmering and slightly reduced, about 2 minutes. Add remaining 2 tablespoons butter, and swirl skillet to melt. Pour sauce over chicken, and sprinkle with parsley. Serve immediately.