Prepare all the vegetables: cut potatoes into 1-inch / 2.5cm cubes, zucchini into 1/2-inch / 1cm slices or half-moons, carrot into 1/4-inch / 5 mm slices, leek into thin slices, celery stalk into 1/4-inch / 5 mm slices, chop the onion.
In a large pot warm up 2 tablespoons of oil, add the onions, leeks and dried thyme (if you're using fresh thyme, add the sprigs at the end of cooking the onions, not at the beginning). Cook over low heat for about 8 minutes. Transfer to a plate.
Add another 2 tablespoons of oil to the pot, turn the heat to high and add the zucchini, potatoes, carrots, and celery. Cook for about 3 minutes (do not stir for the first minute!) until the vegetables are lightly browned.
Add the onion, leek, and thyme back to the pot and pour over 8 cups water (all vegetables should be covered). Cook over low heat for 15 minutes.
Add the green beans, bring to a boil, simmer for 8 minutes.
Add the green peas and cannellini beans (along with the liquid from the can), simmer for 2 more minutes.
Remove the pot from the heat, season the soup with salt and pepper.
Pour the soup into bowls, add separately cooked pasta and a heaped teaspoon of pistou to each bowl. Stir the pistou into the soup just before serving.