Heat oven to 400 degrees F. In turn, put each breast half between two sheets of plastic wrap and pound firmly with a mallet or other object (your fist will work, too), until evenly flattened and less than 1/4 inch thick. Season with salt (not too much since both prosciutto and Parmigiano-Reggiano are salty) and pepper, then layer on a couple of pieces of prosciutto, some Parmigiano-Reggiano, and a good sprinkling of parsley. Roll up and, if necessary, skewer with a toothpick or two (if your chicken is evenly flat and more or less uniform, and you roll tightly, it will self-seal).
Heat an ovenproof skillet, large enough to accommodate the 4 rolls comfortably, over medium-high for 1 or 2 minutes. Add olive oil and 1 tablespoon of the butter and, when the butter foam subsides, add rolls. Brown the rolls on one side, 3 or 4 minutes. Flip them, then immediately put the pan in the oven. Cook until the rolls are done, about 15 minutes (they will be lightly browned and quite firm when done, but its safest to cut into one to be sure the meat is no longer bright pink). Remove rolls to a cutting board and let rest.
Put the skillet back on the stove over medium-high heat. Add wine and cook, stirring and scraping, until the liquid is all but evaporated. Stir in the vinegar and reduce by half, just a couple of minutes, then stir in the remaining 1 tablespoon butter until it melts; taste and adjust seasoning. Cut the rolls into 1-inch slices, arrange on a serving platter, pour the sauce over all, and serve. Makes 4 servings.