In a large mixing bowl, combine the brown sugar, yellow mustard, Worcestershire sauce, chili powder, onion, garlic, and thyme. In a medium-size skillet over medium heat, combine the apple cider vinegar and chicken broth. Stir in the brown sugar and mustard.
Remove the roast from the slow cooker and shred it with two forks. Stir the shredded pork back into the sauce in the slow cooker.
Spread butter on the insides of both halves of hamburger buns. Over medium heat, toast the buns, buttered side down, in a skillet until golden brown. Fill each toasted bun with pork.