1. Put the oil in a pot over medium heat.
  2. When hot, add the onion, carrot, and celery; sprinkle with salt and pepper.
  3. Cook, stirring frequently, until the vegetables soften and begin to darken around the edges, 10 to 15 minutes.
  4. Add the potatoes and sprinkle with more salt and pepper.
  5. Cook, stirring occasionally, until the vegetables are all brown in spots, 5 to 10 minutes.
  6. Add 6 cups water, stirring to scrape up any brown bits from the bottom of the pan.
  7. Add the tomatoes, bring the mixture to a boil, and then lower the heat so the mixture bubbles steadily.
  8. Cook, stirring every now and then, until the tomatoes break down a bit, 15 minutes.
  9. Add the zucchini and greens and raise the heat if necessary to keep the mixture at a steady bubble.
  10. Cook until all the vegetables are very tender, another 10 to 15 minutes.
  11. Taste and adjust the seasoning and serve topped with the cheese and a drizzle of olive oil.