Preheat a smoker or pit to 235 degrees F with red oak. Meanwhile, rub all sides of the meat with the BBQ rub and let sit while the smoker is heating.
Smoke the meat until the internal temperature reaches 155 degrees F, 11 to 12 hours, making sure to use a drip pan under the meat as the drippings will become the au jus. Remove from the heat and let cool for 2 to 3 hours, and then thinly slice into 4-ounce portions.
Melt the butter slowly to clarify, removing the milk solids and reserving the clarified butter. Heat the meat drippings in a saucepan, adding water to reach the preferred consistency. Preheat a skillet, add the clarified butter and the mushrooms, bell peppers and onions and saute until cooked down, sprinkling with salt and black pepper. Add the meat and au jus and mix together, and then lay the cheese on top to melt. Spread the dressing on the buns and top with the meat, vegetables and au jus. Dig in!
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.