Roast beef and au jus: Unwrap the beef and allow it to sit on the counter for 30 minutes to come closer to room temperature.
While the beef rests, combine salt and black pepper in a small bowl to form your dry rub. With one hand (clean hand), sprinkle the beef roast all over with the seasoning. With your other hand (dirty hand), spread and rub the dry rub into the beef to make sure it's stuck all over and there are no spots without seasoning. You can use your clean hand to keep adding more dry rub while your dirty hand stays dirty, rubbing the rub into the meat.
Pre-heat oven to 350 degrees F (177 C).
In a roasting pan, add the carrots, celery, onions, garlic, and the beef broth/stock. Add the roasting rack and then place the beef on the rack. If you don't have a rack, you can place the beef on top of the pile of vegetables.
Roast at 350 degrees F in oven until the internal temperature reaches 135 degrees. This takes my oven almost exactly 1 and a half hours to go from beef internal temperature of 40 F to 135 F. Your oven might vary.
Once the beef hits 135 F remove it from the oven and allow it to rest prior to slicing. I usually let it rest 45 minutes to an hour (If you're not planning to eat the meat immediately, you can refrigerate and slice when it is cold).
Once the beef is sliced or placed into the fridge to cool, strain the liquid from the roasting pan, and reserve the liquid. This is your au jus.
When you are ready to make sandwiches add your au jus to a pan on the stove on medium heat.
When the au jus is slowly simmering, place four sandwich sized piles of beef into the pan with the au jus. This will warm up the beef.
Sandwich assembly: once everything is ready, slice all the rolls you need and add any hot mustard or other condiments to the bottom roll.
Add cheese to the bottom slices of the rolls.
(This step is optional but very good)
Place the bottom slices of the rolls with the cheese under the broiler for 3 to 4 minutes until the cheese is melty.
With tongs, remove a sandwich sized amount of beef from the pot with the au jus and place it onto the cheese on each of the bottom slices of the rolls.
Dip the top slices of each roll into the au jus and place it on top of the meat to form sandwiches. If your pot is too small, you can always spoon a couple of spoons of au jus onto a plate for top-bun-dipping time.
Serve and enjoy.
If you are eating these as leftovers on the second day, add your au jus to a medium sized pot over medium low heat until the au jus is just starting to simmer.
Then you can add enough sliced beef for the number of sandwiches you plan to make to the simmering au jus. After about a minute or two, the meat should be warmed through. Using tongs, remove the meat from the au jus and add it to each sandwich.
Serve the au jus on the side or dip the bread prior to serving, right in the pot with the au jus.