Place chicken broth in a medium saucepan and bring to a simmer; cover and keep warm. Using 2 forks, shred chicken into small pieces; set aside.
In a 4- to 6-quart Dutch oven heat olive oil over medium heat. Add the onion and cook for 8 to 10 minutes or until onions are tender and golden, stirring frequently. Add the garlic and cook 1 minute more, stirring constantly.
Add the pasta to the Dutch oven. Pour warm broth over pasta. Bring to boiling; reduce heat. Cover and simmer for 4 minutes. Add the broccoli and chicken. Cover and cook for 6 minutes more or until pasta is tender. Add pepper.
Divide pasta mixture among 6 shallow bowls. Sprinkle each serving with Parmesan.