Cook the pasta according to the package directions. Reserve 1 ¼ cups of the cooking water. Drain the pasta and return it to the pot.
Meanwhile, using a fork or your fingers, shred the chicken, discarding the skin and bones.
Toss the pasta with the chicken, rosemary, reserved pasta water, ½ cup of the Parmesan, ½ teaspoon salt, and ¼ teaspoon pepper.
Cook, stirring, over medium-low heat, until the sauce has thickened slightly, 2 to 3 minutes. Sprinkle with the remaining ¼ cup of Parmesan.