Sprinkle pork chops with salt and pepper. Heat oil in a large nonstick skillet over medium-high heat. Add the pork and cook until browned on both sides, 1 to 2 minutes per side. Transfer to a plate. Add prosciutto and cook, stirring constantly, until browned, about 1 minute. Transfer to the plate.
Add cider, apples, 1/4 cup broth, vinegar and sage to the pan and bring to a boil, scraping up any browned bits. Reduce heat to a simmer and cook, stirring occasionally, until the apples have softened, 4 to 6 minutes.
Stir together mustard, cornstarch and the remaining 1/4 cup broth in a small bowl. Add to the pan and bring to a simmer, stirring constantly. Return the pork and prosciutto to the pan and simmer, turning the pork to coat, until heated through, 1 to 2 minutes.