Cook pasta according to package instructions, adding broccoli in the last 2 minutes. Drain, reserving 1/2 cup pasta water, then return pasta and broccoli to the pot.
Mix pesto with reserved pasta water in a small bowl. Add chicken and roasted peppers to the pasta, then stir in the pesto mixture. Heat through and stir in Parmesan cheese.
Divide among bowls, top with extra Parmesan if desired, sprinkle with black pepper, and serve immediately.
Notes
Use freshly grated Parmesan for best flavor. Feel free to add more vegetables or adjust pesto to taste.