Prepare the basil sauce: In a food processor fitted with the metal blade, pulse the basil, Parmesan cheese, walnuts, tomato paste, and garlic until puréed. Add the 5 tablespoons of olive oil and pulse until smooth. Season with salt and pepper. Place in a container, cover, and refrigerate until ready to use. (If you float a ¼-inch layer of olive oil over the basil sauce and keep it covered and refrigerated, it will keep for up to 2 weeks.)

Wash and pick over the dried beans and drain. In a medium saucepan of boiling water, cook the beans for 5 minutes. Remove from the heat, cover, and allow the beans to cool in their liquid. Return the cooled beans to the heat, bring to a boil, then simmer for 1 hour or until tender. (If using canned beans, carefully rinse twice in cold water, and drain.)

In a large nonreactive soup kettle, heat the ¼ cup of olive oil over medium heat. Add the tomatoes and cook for 3 minutes, or until they release their juice.

Add the stock and bring to a simmer. Add the carrots and leeks and simmer for 10 minutes. Add the string beans and the cannellini beans and simmer for 3 minutes. Add the zucchini, vermicelli, celery leaves, and saffron and simmer for 5 minutes until the pasta is tender. Stir in 2 tablespoons of the basil sauce and taste for seasoning.

Ladle the soup into warm soup plates and serve with the remaining basil sauce on the side.