In large pot of boiling salted water, cook pasta 3 minutes less than package instructions. Add green beans and cook 3 minutes, then drain.
Meanwhile, in a large saucepan, melt butter over medium-high. Add onion, celery, and carrots and cook until onion is translucent, about 4 minutes. Add flour and cook, stirring, 1 minute. Gradually add broth, stirring constantly, then bring to a boil. Reduce heat and simmer 5 minutes. Stir in chicken and season with salt and pepper.
Stir pasta into chicken mixture.