Make dry rub: In a small bowl, combine brown sugar, paprika, garlic powder, onion powder, black pepper, cayenne if using, and salt. Pat pork shoulder dry with paper towels. Rub spice mixture all over pork, covering all sides. Let sit at room temperature for 30 minutes.
Slow cook pork: Place seasoned pork in slow cooker. Pour chicken broth or apple cider around (not over) the meat. Cover and cook on low for 8 hours or high for 4-5 hours until pork is fall-apart tender and internal temperature reaches 195-205°F.
Make BBQ sauce: While pork cooks, combine ketchup, apple cider vinegar, brown sugar, Worcestershire sauce, mustard, smoked paprika, and garlic powder in a saucepan. Bring to simmer over medium heat, stirring occasionally. Reduce heat to low and simmer for 15-20 minutes until thickened. Season with salt and pepper. Set aside.
Make coleslaw: In a large bowl, combine shredded cabbage and carrots. In a small bowl, whisk together mayonnaise, apple cider vinegar, sugar, celery seed if using, salt, and pepper. Pour dressing over cabbage mixture and toss to coat. Refrigerate until ready to serve.
Shred pork: Transfer cooked pork to large cutting board. Discard any large fat pieces and bone if present. Use two forks to shred meat into bite-sized pieces. Transfer to large bowl and toss with 1-2 cups of BBQ sauce until well coated. Reserve remaining sauce for serving.
Assemble sandwiches: Butter and toast buns in skillet or under broiler until golden. Pile generous amount of pulled pork on bottom bun. Top with coleslaw and pickle slices if desired. Drizzle with additional BBQ sauce. Place top bun and serve immediately with extra sauce on the side.