Combine herbs, nuts, garlic and grated cheese in food processor. While the machine is running, stream in EVOO, about 1/3 cup, to make a pesto. Scrape sauce into a small dish and reserve.
In a large Dutch oven or soup pot heat 2 tablespoons EVOO, 2 turns if the pan, over medium heat. Add chicken and lightly brown, 3-4 minutes. Add onions, celery, carrot, bell pepper and bay leaf, season with salt and pepper, and sauté 4-5 minutes more. Add stock and/or water, and bring to a boil. Stir in green beans and pasta, and keep at a low boil until pasta is cooked to al dente.
Serve soup in shallow bowls and stir in rounded tablespoonfuls of pesto.