01 - Set oven temperature to 325°F (165°C) to prepare for braising.
02 - Pat pork shoulder dry and season all sides evenly with kosher salt and freshly ground black pepper.
03 - Heat olive oil in a large Dutch oven over medium-high heat. Sear pork on all sides until evenly browned, about 3 to 4 minutes per side. Transfer pork to a plate.
04 - Add sliced onion to the pot and sauté for 3 minutes until softened. Add minced garlic and continue to cook for 1 minute.
05 - Stir in tomato paste, smoked paprika, ground cumin, and brown sugar. Cook for 1 minute, then deglaze with apple cider vinegar.
06 - Pour in root beer and chicken broth. Return pork shoulder to pot, nestling into the liquid.
07 - Bring liquid to a simmer. Cover Dutch oven and transfer to the oven. Braise for 2.5 to 3 hours, turning pork halfway through, until meat is fork-tender.
08 - While pork braises, combine mayonnaise, apple cider vinegar, sugar, celery seed, salt, and pepper in a mixing bowl. Whisk until smooth, then toss with coleslaw mix. Chill until ready to serve.
09 - When pork is cooked, remove Dutch oven from oven. Transfer pork to a large bowl. Skim excess fat from braising liquid, then simmer liquid on the stovetop to reduce volume by half, about 10 to 15 minutes.
10 - Shred pork with two forks, discarding excess fat. Return shredded pork to Dutch oven and mix thoroughly with reduced sauce.
11 - Brush split slider buns with melted butter and toast under broiler or in skillet until golden, if desired.
12 - Pile generous portions of sauced pulled pork onto each slider bun, top with tangy slaw, and serve immediately.