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. In a saucepot over medium-low heat, add the pulled pork and toss with the sauce to coat, stirring throughout. Warm, stirring, until hot, about 15 minutes.
To assemble the sandwiches: Stack the bottom brioche buns with some sauced pork and top with the coleslaw followed by the bun tops.
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.