In a 4- to 6-quart slow cooker, combine the beans, sausage, onion, celery, and thyme. Add the broth and stir to combine.
Cover and cook until the beans are tender, on low for 7 to 8 hours or on high for 4 to 5 hours (this will shorten total cooking time).
Twenty minutes before serving, discard the thyme stems, add the collard greens, cover, and cook until the greens are tender, 15 to 20 minutes. Add the vinegar and ½ teaspoon each salt and pepper.
Drizzle with the olive oil and serve with the bread sticks.