Pat pork chops dry with paper towels. Lay 1 slice prosciutto on each chop. Cover chops with plastic wrap and, using meat pounder, pound each until 1/8 inch thick. Dredge cutlets in flour, shaking to remove excess flour.
Heat 2 tablespoons oil in large skillet over medium-high heat until just smoking. Add half of cutlets, prosciutto side down, and cook until light golden brown on first side, about 3 minutes. Flip cutlets and cook until just cooked through, about 1 minute. Transfer to plate, tent with foil, and repeat with remaining oil and cutlets.
Add garlic to empty skillet and cook until fragrant, about 30 seconds. Add broth and simmer, scraping up browned bits with wooden spoon, until reduced by half, about 5 minutes. Add accumulated pork juices back to pan and whisk in lemon juice, sage, and butter. Season with salt and pepper and pour sauce over cutlets. Serve.