Cut the leeks, potato, onion, celery, zucchini, green beans, and carrots into 1/4-inch dice.
In a 6-quart stockpot, combine 3 tablespoons of the olive oil with the water. Add the vegetables and saute over medium-low heat until all the water evaporates. Do not brown the vegetables.
Add the stock and bring to a boil. Cook at a gentle boil for 30 minutes.
Meanwhile, in a food processor fitted with the metal blade, put the tomatoes, basil, garlic, and remaining 3 tablespoons olive oil. Pulse until pureed.
Stir the puree into the cooked soup. Do not let the soup return to a boil. Season, to taste, with salt and pepper. Serve the soup hot or cold from a tureen or in individual bowls.
Cook's Note: My mother uses only water, because her vegetables are peak-of-season, just-picked, and therefore full of flavor; but you can use stock if you like. Pistou, the puree of tomatoes, basil, garlic, and olive oil in this recipe, is a traditional French condiment that adds a burst of flavor just before serving.