Pat pork shoulder dry with paper towels. In a small bowl mix kosher salt, black pepper, paprika, brown sugar, garlic powder and onion powder. Rub the spice mixture all over the pork, pressing to adhere.

Sear (optional, better flavor)

Heat 1 tablespoon vegetable oil in a large heavy pot or Dutch oven over medium-high heat until shimmering. Sear the pork on all sides, 3–4 minutes per side, until browned. Remove pork briefly to a plate.

Braise

Preheat oven to 300°F (150°C). Place the pork back in the pot (or transfer to a roasting pan). Pour 1 cup apple cider vinegar and 1 cup broth into the pot around the pork (not over the spice rub). Cover tightly with a lid or foil. Cook in the oven for 3 to 3½ hours, until the pork is fork-tender and pulls apart easily. Alternatively, cook in a slow cooker on low for 8 hours.

Shred and finish

Remove pork to a cutting board and let rest 10 minutes. Using two forks or meat claws, shred the pork into bite-sized pieces, discarding excess fat and bone. Skim any fat from the cooking liquid, then stir 1 cup barbecue sauce into the shredded pork and add ¼ to ½ cup of cooking liquid to moisten, adjusting to taste. Keep warm until assembly.

Coleslaw

Whisk together mayonnaise, apple cider vinegar, Dijon mustard, honey, celery seed, salt and pepper in a medium bowl until smooth.

Add shredded green cabbage, red cabbage and grated carrot to the bowl. Toss thoroughly to coat. Taste and adjust seasoning — add a little more vinegar if you prefer tangier slaw, or a touch more honey for sweetness. Refrigerate for at least 20 minutes to let flavors meld. Drain any excess liquid before assembling sandwiches if slaw becomes watery.

Panko-crusted onion ring

Prepare onion

Peel the onion and slice into 1/2-inch rings. Separate into individual rings and set aside. Keep smaller inner rings for other uses or make additional rings as desired.

Set up breading station

In a shallow bowl combine flour, 1 teaspoon salt and 1/2 teaspoon pepper. In a second bowl whisk together egg and milk. In a third bowl combine panko breadcrumbs and grated Parmesan (if using).

Bread the rings

Working in batches, dredge each onion ring in flour (shake off excess), dip into the egg-milk mixture, then press into panko so it adheres well. Place breaded rings on a tray and let rest 5 minutes to help coating set.

Fry

Pour 2 cups vegetable oil into a heavy skillet to a depth of about 1/2 inch (or use a deep fryer per manufacturer instructions). Heat oil to 350°F (175°C). Fry onion rings in batches, 1–2 minutes per side, until golden brown and crispy. Do not overcrowd. Transfer to a wire rack or paper towels and season immediately with a pinch of salt. Keep warm in a low oven (200°F/95°C) if needed.

Assembly

Toast buns: spread softened butter on cut sides of 6 buns. Toast in a skillet over medium heat or on a baking sheet in a 375°F oven until golden, 2–3 minutes per side for skillet or 6–8 minutes in oven.

Build sandwiches: Place a generous portion (about 3/4 to 1 cup) of warm pulled pork on the bottom bun. Spoon 1–2 tablespoons additional barbecue sauce over the pork if desired. Top with 2–3 tablespoons coleslaw, then add a panko-crusted onion ring. Add pickles if using, then cap with the top bun.

Serve immediately while pork is warm and onion ring remains crispy. Provide extra barbecue sauce and napkins.