Pat the pork shoulder dry with paper towels. Spread the yellow mustard over the exterior of the pork to help the rub adhere. In a small bowl, combine brown sugar, smoked paprika, kosher salt, black pepper, garlic powder, and onion powder. Rub this spice mixture all over the pork, pressing to adhere.

Braise or slow-cook

Place the seasoned pork in a large Dutch oven or slow cooker. Pour apple cider vinegar and chicken or pork stock around (not over) the pork so the rub remains on top. Add liquid smoke if using. If using an oven/Dutch oven: cover and roast at 300°F (150°C) for 3 to 4 hours, until the meat is fork-tender and pulls apart easily. If using a slow cooker: cook on low for 7 to 8 hours or high for 4 to 5 hours.

Shred and finish

Remove the pork to a cutting board and let rest 10 minutes. Using two forks (or a stand mixer with paddle), shred the pork into bite-sized strands, discarding excess fat. Skim and reserve 1/2 cup of the cooking liquid; mix a few tablespoons into the shredded pork to keep it moist. Stir in 1/2 to 3/4 cup barbecue sauce, adjusting to taste. Keep warm until ready to assemble sandwiches.

Hillbilly Deli Slaw

In a large bowl, whisk together mayonnaise, apple cider vinegar, creamy mustard, honey, celery seed, salt, and black pepper until smooth. Taste and adjust seasoning.

Add shredded cabbage and grated carrot to the dressing. Toss thoroughly to coat. Refrigerate for at least 30 minutes to let flavors meld. Stir again before serving.

Sandwich & Finishes

Toast the buns: spread softened butter on cut faces of the rolls. Heat a skillet or griddle over medium heat and toast buns butter-side-down until golden, about 1–2 minutes. If using cheese, place a slice on the bottom bun while still warm so it melts slightly.

Assemble sandwiches: pile 1/2 cup (about) of warm pulled pork onto the bottom bun. Spoon a tablespoon or two of extra barbecue sauce over the pork if desired. Top with 1/3 to 1/2 cup of Hillbilly Deli Slaw, add pickles to taste, then cap with the top bun. Repeat for remaining sandwiches.

Serve: Serve warm with extra barbecue sauce on the side and kettle chips or fries. Leftover pork keeps in the refrigerator for up to 4 days and reheats gently in a skillet with a splash of reserved cooking liquid.