Heat the oil in a large saucepan or stockpot over medium-high heat. Add the onion, carrots, and celery and cook until softened but not browned, about 5 minutes. Add the broth, 1 cup of water, the potatoes, thyme, and salt. Bring to a boil. Reduce heat, cover partially, and simmer for 15 minutes.
Add the tomatoes, green beans, and broccoli, return to a simmer, and cook until the vegetables are tender, 5 to 10 minutes. Ladle the soup into bowls, sprinkle with the Parmesan (if using), and serve with the baguette slices (if using).
Next Time, Try... Making an improvised minestrone. When you add the potatoes to the pot, also include one 15-ounce can of cannellini beans (drained), 1/2 head of cabbage (shredded), and a handful of macaroni or broken spaghetti.