Combine 1 tablespoon brown sugar, the paprika, mustard powder, cumin, 2 teaspoons salt and ½ teaspoon pepper in a small bowl. Rub the spice mixture all over the pork.
Heat the vegetable oil in a large skillet over medium heat. Add the pork and cook, turning, until browned on all sides, about 5 minutes. Remove the pork and transfer to a plate; whisk ¾ cup water into the drippings in the skillet. Transfer the liquid to a 5- to 6-quart slow cooker.
Add the vinegar, tomato paste, the remaining 2 tablespoons brown sugar and 2 cups water to the slow cooker and whisk to combine. Add the pork, cover and cook on low, 8 hours.
Remove the pork and transfer to a cutting board. Strain the liquid into a saucepan, bring to a boil and cook until reduced by half, about 10 minutes; season with salt. Roughly chop or shred the pork. Transfer to a bowl and toss with 1 cup of the reduced cooking liquid and salt and vinegar to taste. Serve on the buns with barbecue sauce, coleslaw and pickles.
