01 - Preheat oven to 325°F.
02 - Pat the short ribs dry and season with salt and pepper.
03 - Heat vegetable oil in a large Dutch oven over medium-high heat. Sear the short ribs on all sides until deeply browned, about 8 minutes. Remove and set aside.
04 - Lower heat to medium. Add sliced onion and cook until softened, about 5 minutes. Stir in minced garlic and tomato paste; cook for 1 minute.
05 - Pour in beef broth and root beer, scraping any browned bits from the bottom. Stir in Worcestershire sauce, apple cider vinegar, thyme, and bay leaf.
06 - Return short ribs to the pot, submerging in the liquid. Bring to a simmer, then cover and transfer to the oven. Braise for 3 hours, until meat is fork-tender.
07 - While ribs braise, prepare the slaw: In a bowl, combine cabbage, carrot, mayonnaise, apple cider vinegar, and sugar. Season with salt and pepper. Mix well and refrigerate.
08 - When ribs are done, remove from oven. Discard thyme stems and bay leaf. Transfer short ribs to a plate and shred with two forks, discarding excess fat.
09 - Skim fat from braising liquid. Simmer liquid over medium heat until slightly thickened, about 10 minutes. Return shredded beef to the sauce, mixing to coat.
10 - Brush slider buns with melted butter and toast in a skillet or under a broiler until golden.
11 - Place a generous spoonful of braised beef on each bun bottom, top with slaw and pickles if desired, and finish with bun tops. Serve warm.