Heat the oil in a heavy 8 quart soup pot over medium heat.
Add the onion, leeks, and garlic and saute until translucent, about 6 to 8 minutes.
Stir in the carrots, celery, and zucchini and saute until the vegetables are tender, about 10 minutes.
Add the broth and diced tomatoes.
Stir in the basil, parsley, oregano and bay leaves and bring to a boil for 5 minutes.
Reduce the heat, bringing the soup to a simmer and add the green beans, corn and peas.
Cover the pot, continue to simmer for 30 to 45 minutes allowing the flavors to develop while stirring occasionally. If needed, more broth can be added to preserve soup consistency. Season with salt and pepper.
In a separate large pot, cook the cavatappi pasta according to package directions.
When ready to serve, ladle the soup into bowls, add a small portion of pasta and sprinkle with Parmesan cheese.