In a 4- or 5-quart Dutch oven, cook pasta according to package directions, adding vegetables during the last 5 minutes of cooking. Drain. Return to Dutch oven.
While pasta is cooking, in a small bowl, combine Alfredo sauce, pesto, and 1/4 cup milk; set aside. Remove meat from chicken (discard skin and bones); chop or shred meat.
Add chicken to pasta and vegetables in Dutch oven. Add sauce mixture; toss gently to coat. Heat through over medium-low heat. If desired, stir in additional milk to reach desired consistency. Sprinkle each serving with cheese.