Process all ingredients in food processor until smooth, scraping down sides of bowl as needed, about 15 seconds. (Pistou can be refrigerated for up to 4 hours.)
Heat oil in large Dutch oven over medium heat until shimmering. Add leek, celery, carrot, and ½ teaspoon salt and cook 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 pasta and simmer until slightly softened, about 5 minutes.
Stir in haricots verts and simmer until bright green but still crunchy, 3 to 5 minutes.
Stir in cannellini beans and their liquid, zucchini, and tomato and simmer until pasta and vegetables are tender, about 3 minutes.
Season with salt and pepper to taste.
Serve, topping individual portions with generous tablespoon pistou.
