If cutlets are thick, lay them between 2 sheets of wax paper and pound to 1/2-inch thickness using the flat side of a meat mallet or the bottom of a small, heavy skillet. Season with salt and pepper. Place 2 large sage leaves on top of each cutlet, then wrap 2 slices prosciutto crosswise around each cutlet, holding sage in place.

Heat 1 1/2 tablespoons olive oil in a large skillet over medium heat. Add 4 cutlets and cook until no longer pink in the center and the juices run clear, about 2 minutes per side. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Transfer to a platter; tent with foil to keep warm. Repeat with remaining 1 1/2 tablespoons olive oil and remaining 4 cutlets.

Pour wine into the same skillet; simmer, scraping up any browned bits with a wooden spoon, about 1 minute. Pour in broth and simmer until slightly reduced, 4 to 5 minutes. Remove from heat; add butter and swirl the skillet until butter has melted and sauce is creamy. Spoon 2 tablespoons sauce over each cutlet to serve.