In a Dutch oven, heat the oil and sauté onion, carrots, celery, and green beans. Do not brown. Add the rest of the ingredients, reserving the potatoes and the cheese. Bring to a boil and then reduce heat to simmer. Simmer for about 40 minutes. Add the potatoes and continue cooking until potatoes are tender, about 40 minutes longer.
Taste and adjust seasonings, as needed. Ladle into soup bowls and sprinkle with Parmesan.