Preheat oven to 425ºF.
Parboil the asparagus in boiling salted water for 6 to 8 minutes or until tender but not completely cooked. Transfer to a plate and set aside.
Place each chicken breast between 2 pieces of plastic wrap and pound until they are 1/4-inch thick with the flat side of a mallet. Cut the mortadella and fontina slices in half so that they’re slightly smaller than the chicken slices. Season both sides of the breasts with salt and pepper.
Lay a slice of mortadella (2 halves) and then a slice of fontina over each piece of chicken. Place two asparagus spears in the middle of each piece of chicken, with the tips hanging out of the sides. Roll up each chicken breast and secure the ends with two bamboo skewers.
In a 12-inch sauté pan, heat the butter and olive oil and fry the rolls over low heat for about 2 minutes on each side, turning frequently, until they are a light golden color. Transfer to a lined sheet pan and finish cooking in the oven for an additional 10 minutes.
Transfer to a serving plate, add plenty of the mushroom gravy, and sprinkle with fresh parsley. Remove the skewers before serving.