Big Red Slow-Cooker Pulled Pork
['1 tablespoon plus 2 teaspoons kosher salt', '2 teaspoons freshly ground black pepper', '1 1/2 teaspoons ground cumin', '1 1/2 teaspoons smoked paprika', '3/4 teaspoon cayenne pepper', '1 (6-pound) skinless, boneless pork shoulder roast (pork butt or Boston butt)', '1 cup Big Red soda', '1/2 cup store-bought or homemade barbecue sauce (not too sweet)', '1 cup store-bought or homemade barbecue sauce (not too sweet)', '2 tablespoons (or more) Big Red soda', 'Kosher salt, freshly ground pepper (optional)', 'Hamburger buns, sliced pickles, and vinegar-based cabbage slaw (for serving)', 'A large slow-cooker (at least 6 quarts)']
Mix salt, black pepper, cumin, paprika, and cayenne in a small bowl. Rub all over pork.
Stir soda and barbecue sauce in slow cooker. Place pork in slow cooker, cover, and cook on low, spooning juices over top of pork or turning occasionally, until meat is very tender and falls apart when shredded with 2 forks, about 8 hours.
Transfer pork to a rimmed baking sheet and let cool slightly. Shred pork with 2 forks. Discard large pieces of unrendered fat or gristle.
Stir barbecue sauce and 2 Tbsp. soda in a medium bowl. Taste and add up to 1 Tbsp. Big Red, depending on sweetness of barbecue sauce. Toss shredded pork in desired amount of sauce and drizzle more sauce over. You can also spoon some cooking liquid over pork for extra flavor. Season with salt and pepper, if needed.
Serve pork with Big Red sauce alongside, or make sandwiches with buns, pork, slaw, pickles, and an extra drizzle of sauce, if desired.
Pork can be coated in spice rub 1 day ahead; chill. Pork can be cooked and shredded up to 3 days ahead; transfer to an airtight container and chill.
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