Cook pasta according to package directions, adding broccoli the last 2 minutes of cooking. Drain, reserving 1/2 cup of the pasta water. Return drained pasta and broccoli to saucepan.
In a small bowl combine pesto and the reserved 1/2 cup pasta water. Add chicken, roasted peppers, and the pesto mixture to pasta in saucepan. Stir gently to mix. Cook over medium heat until heated through. Stir in the 1/4 cup Parmesan cheese.
Divide the pasta mixture among warm serving bowls. If desired, top with additional Parmesan cheese. Sprinkle with black pepper and serve immediately.