For the pork shoulder: Position a rack in the center of the oven and preheat to 325 degrees F.
Combine the pickle juice and yellow mustard in a small bowl and stir to combine. Combine the brown sugar, chili powder, granulated garlic, onion powder, smoked paprika, chile flakes, 2 teaspoons salt and several grinds of black pepper in a small bowl and stir to combine. Put the pork shoulder into a 13-by-9-inch baking dish. Use your hands to rub the pickle juice and mustard mixture all over the pork followed by the spice mix.
Pour 1/2 cup water and the Worcestershire sauce into the bottom of the dish, stirring to combine with any pickle juice or mustard that made its way to the bottom of the dish. Roast until very tender and a thermometer inserted into the center of the pork shoulder registers 180 degrees F, 3 to 3 1/2 hours, basting the pork with the juices at the bottom of the pan every hour (add water if needed so there is always at least 1/8-inch of liquid in the pan). If the pork starts to get too dark at any point, cover it loosely with foil.
Remove the pork from the oven and use 2 forks to shred the meat into large pieces.
For the smoked beef sausage: Heat the oil in a large saute pan set over medium-high heat until it starts to shimmer. Add the sliced sausage in one layer and cook until the first side is deeply browned in spots, about 2 minutes. Flip the sausage pieces and cook until the second side is deeply browned in spots, about 2 minutes. Use a slotted spoon to transfer the sausage to a bowl.
To assemble: Layer the sausage slices, pulled pork shoulder, a generous spoonful of the onion BBQ sauce and the sliced red onions between the toasted potato hamburger buns.