In a large, heavy-bottomed 10-quart stockpot, combine the olive oil, 3 halved garlic cloves, bouquet garni (several bay leaves and sprigs of thyme wrapped with the green part of a leek and tied securely with cotton twine), and salt. Stir to coat with the oil and cook over moderate heat until the garlic is fragrant and soft, about 2 minutes. Add the fresh or prepared dried beans, and stir to coat. Cook for 1 minute more. Add the leeks, carrots, onions, potatoes, and quartered garlic cloves, and cook until softened over moderate heat, stirring regularly, for about 10 minutes. This will give a lovely color and rich flavor to the soup. Add the zucchini, tomatoes, tomato paste, green beans, 3 quarts cold water, and salt, to taste. Simmer gently, uncovered, until the navy and cranberry beans are tender, about 30 minutes (cooking time will vary according to the freshness of the beans). Add additional water if the soup becomes too thick. Add the pasta, and stir frequently to keep the pasta from sticking to the bottom of the pot, simmering until the pasta is cooked, about 5 minutes more. Taste for seasoning. Remove, and discard the bouquet garni. Serve the soup very hot, passing the Pistou Sauce and cheese to swirl into the soup.
By hand: Place the garlic and salt in a mortar, and mash with a pestle to form a paste. Add the basil, little by little, pounding and turning the pestle with a grinding motion to form a paste. Slowly add the oil, drop by drop, until all the oil has been used and the paste is homogenous. Add salt, to taste. Stir again before serving. In a food processor: Place the minced garlic, salt, and basil in the bowl of a food processor, and process to a paste. Add the oil and process again. Add salt, to taste. Stir again before serving. Transfer to a small bowl. Serve immediately. The sauce can be stored, covered and refrigerated, for 1 day, or frozen for up to 6 months. Bring to room temperature, and stir again before serving. Yield: about 1 cup