1. Combine ham hocks, onion, garlic, celery, thyme, bay leaf, chile, 3/4 teaspoon salt, and the water in a large pot. Bring to a boil over high heat. Reduce heat, and simmer, uncovered, 1 hour 30 minutes.

2. Transfer ham hocks to a plate. Discard chile, and add potatoes to pot; simmer until just tender, 10 to 12 minutes.

3. When ham hocks are cool enough to handle, pull meat off bones (discard bones), shred into bite-size pieces, and return to pot. Add green beans, and return to a simmer. Cover, and cook until beans are just tender, about 8 minutes. Season with pepper. Serve vegetables and meat with a slotted spoon; reserve broth for another use. If storing leftovers, store in broth.