For the pulled pork and Carolina sauce: Preheat the oven to 325 degrees F.
Sprinkle the Boston butt with the Cajun spice, 1 tablespoon salt and 1 teaspoon pepper and put in a deep roasting pan. Add the diced vegetables and bay leaves, and then cover with chicken stock. Wrap with plastic wrap and tightly cover with foil. Roast until tender, 2 to 2 1/2 hours.
Remove the pork from the stock and let cool. Remove the fat from the stock and measure out 1 cup of the stock and cooked vegetable mixture. Let cool slightly, and then put in a blender. Add the mustard, hot sauce, white balsamic, brown sugar, cumin, paprika and some salt and pepper and puree until smooth. Taste the sauce and add more salt and pepper if needed.
Shred the pork with two forks and measure out 2 cups for the sandwiches. Put the pork in a pan and toss with 1/2 cup of the sauce. (Reserve the remaining pulled pork and sauce for another use.) Warm the mixture over medium heat, stirring and tossing, until hot. Lower the heat and keep warm.
For the chicken: Sprinkle the chicken with the salt and pepper. In a large pan over high heat, add the oil. Add the chicken and cook until browned on both sides, 4 to 5 minutes per side.
To assemble the sandwiches: Stack the bottom brioche buns with the chicken, sauced pork and coleslaw, and then finish with the top buns.
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.