Using a food processor and a blender for the potent basil puree gives it an ultra-smooth consistency that helps it blend easily into the vegetable soup.
In a small saucepan, combine the beans with the bay leaf and 2 cups of the water. Cover partially and simmer over low heat until tender, about 20 minutes for fresh beans and 45 minutes for dried. Discard the bay leaf.
Heat 2 tablespoons of the olive oil in a large saucepan. Add the leek, shallot, celery, carrots, bell pepper, thyme and the remaining 3 tablespoons of water and cook over moderate heat until the water has evaporated and the vegetables soften, about 10 minutes.
Add the stock and a large pinch each of salt and pepper; bring to a simmer. Add the green beans, corn and the cooked shell beans along with any remaining cooking liquid. Cover and simmer over low heat until the corn is tender, about 5 minutes.
Meanwhile, in a food processor, combine the basil and garlic and process to a paste. With the machine on, slowly pour in the remaining 1/2 cup of olive oil. Transfer the puree to a blender and blend until very smooth. Scrape the puree into a bowl and season with salt.
Add the zucchini to the soup and simmer just until tender, about 4 minutes. Discard the thyme sprigs and season the soup with salt and pepper. Remove the soup from the heat, stir in the basil puree and serve at once.