Pat the pork shoulder dry very well. Mix the salt, black pepper, garlic powder, onion powder, smoked paprika, cumin, and dried oregano. Rub the seasoning evenly over all sides of the pork. Heat the lard or oil in a heavy pot or Dutch oven over medium-high heat. Sear the pork in batches if needed so the pot is not crowded. Brown the meat deeply on multiple sides, 2–4 minutes per side, until a dark crust forms. Do not rush this stage. The surface should be well browned, not pale, because this is where much of the final savoury depth comes from. Remove the pork to a plate once all pieces are seared. Searing in properly spaced batches, rather than crowding the pot, is what allows real contact between the meat's surface and the hot pot — crowded pork releases moisture and steams rather than browning, and pale, unseared pork produces a noticeably flatter finished sauce regardless of how well the braise itself goes.

Lower the heat slightly. In a bowl, mix the barbecue sauce, water or stock, apple cider vinegar, and Worcestershire sauce if using. Pour this into the pot and scrape up the browned bits from the bottom. Return the pork to the pot. The liquid should come partway up the sides of the meat, not fully cover it. Bring to a gentle simmer, then cover with a lid and cook over very low heat for 2½ to 3 hours, turning once or twice during cooking, until the pork is fully tender and pulls apart easily with a fork. The correct doneness point is when the collagen has broken down completely and the meat shreds without resistance. If it still feels tight or firm, keep cooking and check again in 20-minute intervals. Keeping the liquid at only partway up the meat rather than fully submerging it concentrates the braise's flavour into a smaller volume — a fully submerged braise dilutes both the meat and the eventual sauce, producing a thinner result even after the same amount of later reduction.

Lift the cooked pork out of the pot and transfer it to a tray or bowl. Let it cool just enough to handle. Skim off excess fat from the braising liquid if needed. Shred the pork using your hands once cool enough, or use two forks while still hot. Remove any large hard fat pieces. Keep the shreds moderately chunky rather than too fine so the sandwich keeps texture. Shredding into moderately chunky pieces rather than very fine strands is a deliberate texture choice — overly fine pork loses its individual character and turns mushy once tossed with sauce, while chunkier shreds hold their own texture against the glossy glaze.

Place the pot with the braising liquid over medium heat and simmer uncovered until reduced to a thick, glossy sauce consistency, about 8–15 minutes. Stir occasionally so the sugars do not catch on the bottom. The sauce should coat a spoon and no longer look thin or watery. Taste and adjust. Add the brown sugar only if the sauce needs a touch more sweetness or roundness. Add a pinch more salt or pepper if needed. Return the shredded pork to the reduced sauce and toss until evenly coated. The finished meat should be glossy, juicy, and smoky, with enough sauce to cling to every strand but not pool heavily in the bottom of the bowl. Reducing the braising liquid as a separate step, after the pork is already out, concentrates its flavour and thickens it into something that clings rather than pools — tossing the pork back into an unreduced, thin liquid instead would leave the meat wet and leak straight through the bun.

While the pork finishes, combine the cabbage and carrot in a bowl. In a separate small bowl, mix the mayonnaise, sour cream, Dijon mustard, apple cider vinegar or lemon juice, sugar, a small pinch of salt, and a few grinds of black pepper. Pour over the vegetables and toss until evenly coated. The slaw should be lightly creamy, fresh, and crisp, not overdressed. Let it sit 10–15 minutes so the cabbage softens slightly but still keeps crunch. This brief rest softens the cabbage's rawest edge just enough to make it pleasant to eat while still preserving genuine crunch — a longer sit would push the slaw past that point into something limper and wetter.

Split the brioche buns. Lightly butter the cut sides and toast them in a pan or on a griddle over medium heat until golden. Pile a generous portion of the warm pulled pork onto each bottom bun. Top with a mound of fresh coleslaw and close with the top bun. Serve immediately while the pork is hot and glossy and the slaw is still crisp. Toasting the bun gives it real structural resistance against the warm, glossy pork's moisture, and building the sandwich with hot pork first, cool slaw last, keeps that fresh topping from wilting under residual heat before serving.