COOKING INSTRUCTIONS: Place chicken between sheets of heavy-duty plastic wrap and pound to flatten to cutlets. Place a sage leaf (or two) and a slice (or two) of prosciutto on top, pressing prosciutto to adhere to the cutlets and trimming to fit, if necessary.
Place the Parmesan cheese in a shallow dish; dredge chicken until well coated on both sides. Heat 1 tablespoon of butter and the oil in a large skillet over medium-high heat until butter begins to foam. Add chicken cutlets, prosciutto sides up; cook until chicken juices run clear and lightly browned. Turn off the heat then transfer chicken to a platter and cover to keep warm, but do not remove the pan from the stove.
Place the pan over medium-high heat; add wine (or vermouth) and whisk up all of the browned bits from the bottom of the pan. Continue to cook until mixture is reduced by half. Add broth, return mixture to a boil and cook again until reduced by half.
Remove pan from heat and swirl in remaining butter; check for seasoning then pour sauce over chicken cutlets. Garnish each serving with a lemon wedge.