Pound the chicken cutlets to about ½-inch thickness. Season both sides with salt and pepper, then lightly dredge in flour (shake off excess).
Heat olive oil in a large skillet over medium-high heat. Cook chicken 3–4 minutes per side until golden and cooked through. Transfer to a plate.
In the same pan, lower heat to medium. Add 2 tbsp butter and garlic; sauté for about 30 seconds. Pour in white wine and scrape up browned bits (this is flavor gold). Let it reduce by half.
Stir in lemon juice, chicken broth, and capers. Simmer for 3–5 minutes until slightly thickened.
Return chicken to the pan. Add remaining butter and spoon sauce over the top. Simmer 2 minutes until everything is glossy and coated.
Garnish with parsley and extra lemon slices.
