For the pistou: Process all ingredients in food processor until smooth, about 15 seconds, scraping down sides of bowl as needed. Pistou can be refrigerated for up to 4 hours.
For the soup: Heat oil in Dutch oven over medium heat until shimmering; add leek, celery, carrot, and ½ teaspoon salt and cook, stirring occasionally, until vegetables are softened, 8 to 10 minutes. Stir in garlic and cook until fragrant, about 30 seconds. Stir in broth and water and bring to simmer.
Stir in peas and simmer until slightly softened, about 5 minutes. Stir in haricots verts and simmer until bright green but still crunchy, about 1 minute. Stir in cannellini beans, zucchini, and tomato and simmer until peas and vegetables are tender, about 1 minute. Season with salt and pepper to taste.
Serve, topping individual portions with pistou.
