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Layer the ingredients in the slow cooker: Place the smoked ham hocks in the bottom of a large slow cooker (5–7 quarts). Scatter the quartered yellow potatoes over the ham, then add the trimmed green beans on top.
Pour in the broth: Slowly pour the chicken broth over everything in the slow cooker. The liquid should mostly cover the potatoes and beans; it’s okay if a few pieces peek out, as they’ll cook down.
Cover and cook: Put the lid on the slow cooker and cook on LOW for 7 to 8 hours, or on HIGH for 4 to 5 hours, until the ham is very tender, the green beans are soft, and the potatoes are easily pierced with a fork.
Shred the ham: Carefully transfer the ham hocks to a cutting board or plate. Remove and discard the skin, bones, and any large fatty pieces. Shred or chop the remaining ham into chunky bite-size pieces.
Return ham to the pot: Add the shredded ham back into the slow cooker and stir gently to combine it with the green beans, potatoes, and broth. Taste the broth; if your ham wasn’t very salty, you can add a pinch of salt and a little black pepper at this point if desired.
Serve: Ladle the ham, green beans, and potatoes into bowls, making sure to spoon plenty of the clear, savory broth over each serving. Serve hot, straight from the slow cooker, with extra black pepper on top if you like.