To prepare soup, heat a large saucepan over medium-high heat. Add 1 teaspoon oil. Add leek and a pinch of salt, and sauté 3 minutes, or until very tender. Toss in carrot and celery, and saute another 4 minutes, or until tender. Add garlic, and saute 30 seconds, or until just fragrant.
Add water, tomatoes, potatoes, a pinch of salt, thyme sprigs and bay leaf, and bring to a boil. Reduce heat, and simmer 10 minutes, or until potatoes are tender. Add snap peas, and cook 4 minutes, or until crisp-tender. Add beans, and cook 2 minutes. Season to taste with black pepper.
While soup simmers, prepare pistou. If using a mortar and pestle, place garlic in the bowl with a pinch of salt, and pound to a paste. Then add basil and mint a few leaves at a time, and continue to pound to a paste until herbs are gone. Whisk in 2 tablespoons olive oil and cheese, and season with pepper to taste. If using a blender or mini food processor, add garlic to bowl with a pinch of salt, and pulse until minced. Add basil and mint, and pulse a few times until herbs are chopped. Drizzle in oil, and process until herbs are very finely chopped. Add cheese, and pulse just until combined. Serve with soup.