Start by dicing the potatoes, slicing the zucchini, carrots, leek, and celery, and chopping the onion. Aim for uniform pieces to ensure even cooking.

In a large pot, heat 2 tablespoons of olive oil over low heat. Add the chopped onions and leeks, cooking them gently for about 8 minutes until they become softened and translucent.

Increase the heat to medium and add another 2 tablespoons of olive oil to the pot. Toss in the zucchini, diced potatoes, carrots, and celery, cooking for 3 minutes without stirring initially.

Next, incorporate the sautéed onions and leeks into the pot along with 8 cups of water. Add the dried thyme, stir everything together, and bring the mixture to a low boil.

Raise the heat to bring the pot back to a boil, then add the green beans. Allow them to cook for 8 minutes, maintaining a lively simmer.

Toss in the green peas and cannellini beans into the simmering soup. Stir well and let everything simmer for another 2 minutes.

In a food processor, combine the fresh basil leaves, garlic, and grated Parmesan cheese to create a chunky paste. Drizzle in olive oil until smooth.

To serve, ladle the soup into bowls and add a small amount of cooked elbow pasta, if desired. Top each bowl with a generous teaspoon of the pistou.