For the brussels sprouts: Cut them in half and then shred them fine with a big, sharp knife. If you have a few shallots, sliced those up too. Heat a skillet over high heat. Add a bit of the oil of your choosing. Throw the shredded greens (and shallots) in the hot pan with a big pinch of salt. Give them a stir and let them really brown. Add a glug of apple juice, turn the heat down to low, and braise them in the juice until they're soft and sweet, about 15 minutes. When the apple juice has cooked off, and your brussels sprouts are tender, season them with a splash of apple cider vinegar, a few cracks of black pepper, and a pinch more salt if it needs it.
As per usual, this Saturday sandwich is totally designed to go with a beer.