Apply mustard on chuck roast as a binder; season with BBQ rub.
Preheat smoker to 250 degrees with hickory wood chunks.
Smoke chuck roast, spritzing with apple cider vinegar every 30-40 minutes until it reaches 165 degrees (about 3 hours).
Place chopped onions and minced garlic in an aluminum pan; add smoked chuck roast, beef broth, and additional apple cider vinegar. Cover with foil and return to smoker until meat is tender.
Rest meat for 30-45 minutes; toast King's Hawaiian rolls.
Shred the beef; assemble sliders with pepper jack cheese, shredded beef, BBQ sauce, and cheddar cheese. Cap with the top half of the rolls.
Bake at 415 degrees for 12-15 minutes. Serve with pan liquid as a dipping sauce.