Preheat oven to 250 degrees F (120 degrees C).

Mix brown sugar, paprika, garlic powder, black pepper, and salt in a bowl. Spread mustard over pork roast and sprinkle with brown sugar mixture, using the entire amount. Line a shallow baking dish with aluminum foil. Place a rack in the prepared baking dish and arrange pork roast on rack.

Bake in the preheated oven until very tender, 9 to 11 hours. Let pork cool, then shred into bite-size pieces. Stir barbeque sauce into pork to moisten; serve with hamburger buns.