Preheat your oven to 300°F (150°C) or set a slow cooker to LOW. Pat the Pork Shoulder dry with paper towels.
Make the dry rub by combining Brown Sugar, Paprika, Kosher Salt, Black Pepper, Garlic Powder, Onion Powder, Chili Powder, Ground Cumin and Dry Mustard; rub it all over the pork, pressing it into the surface.
If you like a browned exterior, sear the pork in a hot Dutch oven for 3–4 minutes per side until nicely caramelized, then remove briefly.
Return the pork to the pot (or place in the slow cooker) and add about 1 to 2 cups of Chicken Broth, 1/4 to 1/2 cup of Apple Cider Vinegar, 1–2 tablespoons Worcestershire Sauce, and a splash of Liquid Smoke if using. The liquid should come up the sides but not cover the pork completely.
Cover and braise in the oven for 3–4 hours, or cook in the slow cooker on LOW for 8 hours, until the pork is fork-tender and pulls apart easily.
Transfer the pork to a cutting board or large bowl and shred with two forks, discarding any large pieces of fat. Skim and discard excess cooking fat from the braising liquid, then stir some of the strained juices back into the shredded meat for moisture and flavor.
Stir in Barbecue Sauce to taste, warming the mixture gently on the stove if desired so the sauce melds with the pork.
While the pork finishes, make the slaw: thinly slice or shred the Green Cabbage and grate the Carrot. In a bowl whisk together Mayonnaise, a splash of the same apple cider vinegar from earlier (or additional if needed), Granulated Sugar, Celery Seed, and a pinch of Kosher Salt and Black Pepper to taste. Toss the cabbage and carrot with the dressing and refrigerate briefly to meld.
Lightly toast the Soft Sandwich Bun if you like, pile on a generous portion of the sauced pulled pork, top with a scoop of the creamy slaw, and finish with a few Dill Pickle Chips. Serve warm and enjoy.
