Combine the mustard, sugar, brown sugar, vinegar, chili powder, smoked paprika, white pepper, black pepper, cayenne, soy sauce, butter, liquid smoke, and 1/4 cup water in a large skillet. Bring to a simmer, whisking until smooth, over medium heat. Reduce the heat to low and simmer, stirring occasionally, for 10 minutes.
Meanwhile, discard the skin from the chicken, then shred all of the meat into bite-sized pieces. You should have about 3 1/4 cups meat. Discard the bones. Add the meat to the sauce and fold gently to combine. Remove from the heat.
Place a slice of tomato on each bottom roll, then top with the chicken mixture. Top with the celery and then the tops of the rolls. Serve immediately.