Heat oil in a large saucepan or stockpot over medium-high heat. Add the onion, carrots, and celery. Cook until tender but not brown, about 5 minutes.
Add the broth, 1 cup water, potatoes, thyme, and ½ teaspoon salt. Increase heat to high and bring to a boil.
Reduce heat to medium and simmer, partially covered, 15 minutes.
Add the tomatoes, beans, and broccoli. Cook 5 to 10 minutes more or until all the vegetables are tender.
Sprinkle with Parmesan. Serve with toasted baguette slices (if desired).