Split each chicken breast in to two pieces horizontally and place between 2 sheets of plastic wrap. Using the flat side of a mallet gently pound chicken into ¼ inch thick cutlets.
Season cutlets with salt and pepper, then dredge in flour. Heat oil in a saute pan over medium-high heat. Saute cutlets 2-3 minutes on each side until golden brown and cooked through.
Transfer cutlets to a warmed platter. Deglaze pan with wine. Add minced garlic and cook until garlic is slightly brown and liquid is nearly evaporated, about 2 minutes. Add broth, lemon juice, and capers.
Return cutlets to pan and cook on each side for 1 minute. Transfer cutlets to warm plates.
Finish sauce with butter and lemons. Once butter melts, pour sauce over cutlets. Garnish with chopped fresh parsley and serve immediately.