Cut the pork shoulder into 3 large pieces. Mix the brown sugar, smoked paprika, salt, garlic powder, onion powder, black pepper and cayenne, then coat every surface of the meat.

Set the pork on a rack over a tray and refrigerate uncovered overnight if you can, or leave it 1 hour at room temperature. The dry surface browns instead of steaming.

Heat the oven to 300F. Scatter the sliced yellow onion in a Dutch oven, set the pork on top, and pour the chicken stock and 2 tablespoons cider vinegar around it. The liquid should reach about a third of the way up the meat.

Cover and braise 4 hours, until a fork twisted into the meat turns freely with no resistance. If it resists, give it another 30 minutes.

Move the pork to a board and rest 20 minutes. Strain the braising liquid into a tall measuring cup and let it settle 10 minutes, then spoon off the pale fat layer that rises to the top.

Shred the pork with two forks, pulling with the grain and then across it. Discard any large fat seams.

Stir the barbecue sauce and ½ cup of the defatted braising liquid through the shredded pork until it is glossy and loose, not swimming.

Warm the cider vinegar with the sugar and salt until dissolved, pour over the sliced red onion in a jar, and leave at least 10 minutes.

Raise the oven to 425F. Prick the sweet potatoes, rub the skins with olive oil and salt, and bake unwrapped on a foil-lined sheet pan for 45 to 55 minutes, until a knife meets no resistance.

Split each potato lengthwise, press the ends together to open it, and work ½ tablespoon of butter and a pinch of salt into the hot flesh with a fork.

Fill each potato with the sauced pork, top with drained pickled red onions and sliced green onions, and serve.