Place the ham bone, carrots (cut into chunks), celery stalk (cut into chunks), 2 garlic cloves (unpeeled), thyme sprigs, dried white beans (no soaking needed), and water into the slow cooker.
Cover and cook on low for 8 hours or on high for 4 hours.
Remove the ham bone from the slow cooker. Shred any meat off the bone and discard the bone. Return about half of the shredded ham back into the slow cooker.
Discard the thyme sprigs.
Chop the carrot and celery chunks removed earlier and return them to the slow cooker.
Melt 1 tbsp butter in a skillet over medium-high heat. Add about half of the remaining shredded ham and cook until golden and crispy.
Transfer most of the crispy ham to the slow cooker, reserving a few pieces for garnish.