Using a fork, mash together garlic, mustard, sugar, salt, black pepper, smoked paprika, and cayenne pepper in a small bowl to form a paste.

Spread paste evenly over top of pork. Refrigerate in a roasting pan for at least 1 hour or up to overnight.

Preheat oven to 300 degrees. Add water to bottom of pan, and cover with foil. Cook, basting hourly, flipping halfway through, until pork is tender and pulls apart easily, about 6 hours. Let cool slightly. Using 2 forks, shred pork, and toss with barbecue sauce. Sandwich pork mixture between buns, topping the meat with pickled vegetables.