Heat oven to 350° F. Place 2 sage leaves in the center of each slice of prosciutto, then wrap each cutlet in the prosciutto.
Put the flour and pepper on a plate and dredge the prepared cutlets. Heat the oil and 1 tablespoon of the butter in a large skillet over medium-high heat. Once the butter is foaming, add 4 of the cutlets, seam-side down.
Cook the cutlets until golden brown, about 3 minutes per side. Transfer to a baking dish. Repeat with 1 tablespoon of the butter and the remaining cutlets.
Transfer the baking dish to oven and cook the cutlets for 5 minutes.
Meanwhile, reduce heat under the skillet to medium-low and add the Marsala. Simmer for 3 minutes, stirring constantly and scraping up browned bits.
Add the capers and parsley. Stir in the remaining tablespoon of butter. Spoon the sauce over the chicken.