Cook the pasta according to the package directions.
Shred the chicken, using a fork or your fingers, while the pasta cooks. Discard the skin and bones.
Drain the pasta, reserving 1 1/4 cups of the water. Return the pasta to the pot.
Add the reserved pasta water, chicken, rosemary, 1/2 cup of the Parmesan, 1/2 teaspoon salt, and 1/4 teaspoon pepper to the pot. Stir over medium-low heat until the sauce has thickened slightly, about 2 minutes.
Divide among individual bowls and sprinkle with the remaining Parmesan.
Tip: Using water reserved from the pasta pot is a classic cook's trick. Starchy and salted, it makes a light, flavorful sauce when tossed with cheese and herbs. It's also great for thinning pestos and marinaras.