Mix the brown sugar, chili powder, cumin, mustard powder, allspice, cayenne, 1 teaspoon salt and 1/2 teaspoon black pepper in a bowl. Rub half of the mixture all over the pork; transfer to a large resealable plastic bag and refrigerate 1 to 4 hours.
Preheat the oven to 350 degrees F. Heat the vegetable oil in a large Dutch oven over medium heat. Add the pork and cook, turning occasionally, until browned on all sides, about 8 minutes; remove to a plate.
Add the onion, garlic, 1 teaspoon salt and a few grinds of black pepper to the pot; cook, stirring, until the onion is softened, 5 minutes. Add the tomato paste and the remaining spice mixture; cook, stirring, 2 more minutes. Stir in the chicken broth, ketchup, vinegar and 2 cups water. Tear 1 bun into pieces and add to the pot. Return the pork to the pot and bring to a simmer.
Cover the pot and transfer to the oven. Cook until the pork is very tender, about 2 hours. Uncover and continue cooking until the sauce thickens slightly, about 20 more minutes. Shred the pork with 2 forks, tossing to coat in the sauce. Serve on buns with coleslaw and pickles.