Pat pork dry. Mix together salt, pepper, paprika, garlic powder, onion powder, and brown sugar. Rub all over the pork. Let rest at room temperature 10 to 15 minutes.
Heat olive oil in a large skillet over medium-high heat. Sear pork on all sides until deeply browned, about 3 minutes per side. Transfer to a plate.
Add 1/2 cup broth to the hot skillet. Scrape up all browned bits and simmer 1 to 2 minutes. Remove from heat.
Place onion wedges and smashed garlic in the bottom of the slow cooker. Set seared pork on top.
Whisk together remaining 1 cup broth, Worcestershire sauce, tomato paste, apple cider vinegar, soy sauce, salt, and the deglazing liquid. Pour around the pork, not over it.
Cover and cook on HIGH for 5 to 6 hours until pork is falling-apart tender. Begin checking at 5 hours.
Transfer pork to a cutting board, tent with foil, and rest 10 minutes. Remove onions with a slotted spoon and reserve. Pull pork into bite-sized chunks, discarding large fat pieces.
Skim fat from the cooking liquid. Transfer to a saucepan and simmer uncovered 10 to 12 minutes until slightly reduced. Stir in butter. Adjust seasoning and keep warm.
Arrange split rolls cut-side up on a baking sheet. Broil 1 to 2 minutes until lightly golden.
Spoon pulled pork onto each bun bottom. Top with reserved onions and a slice of cheese. Return to broiler 60 to 90 seconds until cheese is melted and bubbly. Cap with bun tops.
Serve immediately with warm dipping jus on the side.