Place beans in a large container and cover with several inches of cool water; let stand 8 hours to overnight. Drain and rinse.
Heat olive oil in a stockpot over medium heat; cook and stir onion until golden brown, 15 to 20 minutes. Add carrots, mushrooms, celery, kale, chard, and garlic; cook and stir until carrots and mushrooms are slightly tender, 5 to 10 minutes.
Pour broth over vegetable mixture and bring to a boil; reduce heat and add bay leaves, rosemary, basil, salt, and pepper. Add beans and simmer soup until beans are softened, 1 1/2 to 2 hours.
