Season the chuck roast: Pat the chuck roast dry and trim away only any large or loose pieces of exterior fat. Lightly coat the entire roast with hot sauce to use as a binder.
Make the dry rub: Combine the kosher salt, cumin, Mexican oregano, black pepper, garlic powder, cinnamon, cloves, and ginger. Reserve 2 teaspoons of the finished rub for the consommé. Coat the chuck roast evenly with the remaining rub. For the best flavor, refrigerate the seasoned roast overnight, or let it sit for at least 2 hours if making it the same day.
Prepare the dried chiles: Remove the stems and seeds from the ancho, guajillo, and arbol chiles.
Toast and soften the chiles: Add the olive oil and dried chiles to a large skillet or shallow pan over medium heat. Cook for about 5 minutes, stirring frequently, until the chiles become fragrant and soften slightly. Do not let them burn.
Add the stock: Carefully pour in the beef stock. Increase the heat to medium-high and simmer for about 5 minutes.
Blend the consommé: Remove the chile mixture from the heat and let it cool slightly. Add the chiles and stock to a blender along with the chopped onion, garlic, fire-roasted tomatoes, apple cider vinegar, and reserved 2 teaspoons of dry rub. Blend until smooth.
Strain: Pour the blended mixture through a fine-mesh strainer into a large bowl or container. Press the mixture through with a spoon or whisk to extract as much liquid as possible. Discard the solids and set the consommé aside.
Smoke the chuck roast: Preheat the smoker to 275°F for indirect cooking. Place the seasoned chuck roast directly on the smoker grates. Smoke until the internal temperature reaches approximately 165°F.
Braise the beef: Heat the prepared consommé until warm. Transfer the chuck roast to a large foil pan and pour the warm consommé around the roast. Cover the pan tightly with aluminum foil.
Finish cooking: Return the covered pan to the smoker at 275°F. Continue cooking until the chuck roast reaches approximately 203°F and is completely tender. The beef should be probe tender and pull apart easily. If the roast still feels firm, continue cooking until tender rather than relying on temperature alone.
Rest the beef: Remove the covered pan from the smoker and let the beef rest for at least 2 hours.
Shred the birria: Remove the chuck roast from the consommé and shred the beef into bite-sized pieces. Keep some of the shredded beef in the consommé to help retain moisture.
Preheat the oven: Heat the oven to 350°F.
Assemble the sandwiches: Open the French bread rolls and add Oaxaca cheese across both sides of each roll. Pile the shredded birria beef toward the center of the roll, then add another layer of Oaxaca cheese over the beef.
Melt the cheese: Place the sandwiches on a baking sheet and bake for 5 to 6 minutes, or until the cheese is melted and the sandwich is heated through.
Garnish: Remove from the oven and top each sandwich with finely diced white onion and minced cilantro.
Serve: Serve the sandwiches immediately with warm birria consommé in small bowls on the side for dipping.