Add the carrots, celery, and onion, and sauté until slightly cooked.
Add the ham and thyme and continue to cook until the ham is slightly browned.
Add the water and deglaze any fond that's developed.
Add the chicken stock and bring to a boil, then reduce to a simmer and cook, uncovered, until the vegetables are barely tender.
Add the two cans of beans and stir to combine. Bring to a boil, then reduce to a simmer and cook until the vegetables are tender. Serve with cornbread or a hearty bread.
Notes
You can also make this in a small slow cooker. Omit the oil and ¼ cup water, and reduce the chicken stock to ½ a cup. Add everything to a 2-quart slow cooker, place the lid on and cook on high for an hour. Reduce to low and cook for another two to three hours until the vegetables are tender. Because this is low and slow cooking, add a bay leaf at the beginning and remove it before serving.