Strip the tough stems from the greens, tear the leaves into large pieces and wash them in two changes of cold water until no grit settles in the bowl.
Heat the oil in a Dutch oven over medium heat and cook the ham until browned at the edges, 4 to 5 minutes.
Add the onion and cook until soft, 5 minutes, then stir in the garlic for 30 seconds.
Add the greens by handfuls, turning each addition until it wilts enough to make room for the next.
Pour in the broth and add the brown sugar, pepper flakes and salt; bring to a boil, then cover and lower the heat.
Simmer until the greens are silky and no fibrous chew remains, 55 to 65 minutes.
Stir in the vinegar and serve the greens in bowls with some of the smoky broth.