Make the pesto: Put all of the pesto ingredients into a food processor and process until smooth. If needed, add more chicken broth - but not too much. You want it to be a paste.

Cut chicken into bite-sized pieces. Heat the oil in a large skillet over medium heat. Add the chicken, mushrooms and onion and cook until the chicken is cooked through and no longer pink. Add the pasta to the chicken mixture, then stir in the pesto.