Heat olive oil in a large soup pot over medium heat; add leeks and cook and stir until tender and translucent, 3 to 5 minutes. Add garlic; cook and stir until fragrant, about 30 seconds. Add carrots, potatoes, zucchini, green beans, tomatoes with liquid, thyme leaves, bay leaf, water and salt; stir to combine. Being to boil, then reduce heat and simmer 30 to 45 minutes, until vegetables reach your desired tenderness.
Stir in cooked beans and pasta; simmer 10 to 12 minutes more until pasta is tender. Remove the bay leaf and taste for seasoning; add more salt as desired.
For the Pistou
Meanwhile, make the pistou: with food processor running add garlic clove through the chute to mince. Add basil, parmesan cheese and salt; process until minced. With food processor running, drizzle in olive oil in a steady stream and process until combined. Taste for seasoning and add more salt as desired. (recipe note #3)Transfer pistou to a small bowl for serving. (recipe note #4)
To Serve
Ladle soup into bowls, topping each with a large scoop of pistou and serve.
Notes
This makes a very thick soup - particularly any leftovers that spend a night or longer in the refrigerator. Add water (or broth if you prefer) and additional seasoning if you like, to achieve your desired texture and flavor.
When adding salt, remember that the pistou you'll add to finish each bowl is also salty.
Instead of canned beans, you can use 3 cups of dry beans that you've cooked yourself (here's how to do it easily in a slow cooker). Or, even better, fresh shell beans. You'll need 2 pounds for this recipe. Once you remove the fresh beans from the pod, cover with enough water just to cover, bring to boil and add a generous pinch of salt. Reduce heat and simmer 25-30 minutes until just tender. Remove from heat. Drain before adding to the soup.
Or combine ingredients with a mortar and pestle.
If making the pistou well in advance, add to an airtight container. Smooth the top and cover with a thin layer of olive oil to protect the color. Then cover and refrigerate for up to 1 week.