Tip:If the handle of your skillet isn't ovenproof, wrap the handle well with aluminum foil.
Heat the oven to 450°F. Heat a 12-inch ovenproof skillet over medium-high heat. Once it’s hot, add 1-1/2 Tbs. of the oil and then the asparagus. Season with salt and pepper and sauté, shaking the pan often, until the asparagus is slightly tender, 2 to 4 minutes. Transfer the asparagus to a plate and remove the pan from the heat.
Lay the chicken on a cutting board and make a horizontal cut in each, starting on the thickest side of the breast and cutting almost completely through to the other. Open up the breast like a book. Pound with a meat mallet or a small heavy skillet to flatten. Repeat with the remaining breasts. Over both parts of each breast, arrange an even layer of the basil, prosciutto, Parmigiano, and a small handful of the cooked asparagus. Turn each breast so a long side is facing you. Roll the chicken away from you, folding it tightly over itself. Use two toothpicks to seal the rolled area. Season with salt and pepper.
Put the skillet back on medium-high heat, and once it’s very hot, add the remaining 2 Tbs. oil and the chicken rolls. Cook the chicken on both sides until well browned, 3 minutes per side. Add 2 Tbs. water and the balsamic vinegar and scrape the pan to incorporate any browned bits. Partially cover the skillet, put it in the oven, and roast until the chicken is cooked through (an instant-read thermometer should read 160°F), 12 to 15 minutes. Let the chicken rest for a couple of minutes, cut the rolls diagonally in half or slice thinly, and serve with a spoonful of the balsamic sauce,.