Step 1

In a 6- to 8-quart slow cooker, combine the soaked beans, garlic, carrots, celery, Parmesan rind, rosemary, broth, oregano, and garlic and onion powders. Cook on low until the beans are very tender, 6 to 7 hours.

Step 2

Increase the heat to high. Whisk in the tomato sauce, ditalini, tomato paste, and green beans, if using. Season with 2 teaspoons salt and several generous grinds of black pepper. Cover and cook until the pasta and green beans are just tender, 20 to 25 minutes. Remove and discard the rosemary stem and Parmesan rind. Taste and add more salt and pepper, if you like. Serve with grated Parmesan on top, if desired.