Place each piece of chicken between 2 sheets of waxed paper and pound them thin with a mallet, the flat of a large knife, or a rolling pin. Trim the rind from the cheese and cut it into 2 thick sticks.
Wrap each stick with a slice of prosciutto and place in the center of each flattened chicken breast. Roll the chicken breast around the prosciutto and cheese to make a sausage shape and secure with toothpicks or butcher's twine. Season the rolls with salt and pepper.
Heat 2 tablespoons of the butter and 1 tablespoon of the olive oil in a heavy skillet. Quickly brown the chicken rolls over medium-high heat, reduce the heat to low, and scatter the mushrooms into the pan, sprinkle with wine and rosemary. Cook, turning the chicken rolls frequently, 10 to 15 minutes.
When the first bit of cheese begins to seep out, the chicken is done. Remove it from the pan to a cutting board at once. Cut each of the rolls at an angle into 6 slices. Cover to keep warm. Reheat the pan juices, swirl in the remaining 2 tablespoons butter, and season, to taste, with salt and pepper.
To serve arrange the chicken and mushrooms on a bed of arugula leaves tossed with olive oil and seasoned with salt and pepper. Warm the pan juices and strain them over the chicken.