Preheat a cast iron dutch Oven over high heat.
Prep all of the vegetables. Cut both ends off of the green beans, slice the potatoes in half add them to a bowl of cold water to keep their color, and dice the onions.
Using sharp kitchen shears, cut the bacon into pieces right over the Dutch Oven and allow the bacon to start crisping up.
Add the diced onions to the fried bacon pieces and sauté until the onions are fragrant and softening.
Add the garlic to the bacon and onions, sautéing for another 30 seconds until the garlic is also fragrant.
Add the Ham Hocks and evenly spread the green beans through the pot.
Pour in the chicken stock and bring the pot to a boil.
Once it's boiling, reduce the heat to medium-low, cover the pot, and simmer for about 40 minutes.
At the 40-minute point, remove the ham hocks and carefully remove any remaining meat still on the bone. Discard the bones and toss the meat in with the green beans.
Add the potatoes with the green beans and ham, stirring to be sure they are even distributed.
Add the butter, cover the pot, and simmer for 15 minutes, checking the potatoes halfway through with a fork to prevent overcooking.
Turn the heat off and let the pot of green beans and potatoes rest, and the flavors meld.