Heat oil in Dutch oven or soup pot over medium; sauté onion, carrots, and celery 8-10 minutes until softened.
Cook garlic 1 minute, stirring frequently, until aromatic.
Mix in broth, tomatoes, rosemary, thyme, and bay.
Add beans; boil, reduce heat, and simmer 20 minutes to blend flavors.
For creamier texture, partly mash about half the beans against pot side with masher or immersion blender.
Cook kale or spinach 5 minutes until wilted; season with salt and pepper and remove bay.
Ladle into bowls; top with fresh Parmesan and oil drizzle.