Start by applying mustard over the entire chuck roast for the binder. Next season on all side with an all purpose rub.
Preheat the grill to 250 degrees Fahrenheit and set for indirect heat.
Place the seasoned chuck roast directly on the grates and smoke for 3 hours.
Next, transfer the smoked beef into a dutch oven and add all the braise ingredients.
Place the dutch oven on the grill for another 3-4 hours or until the chuck roast is fork tender. Bump up the grill temperature to 300 degrees to speed up the process.
When it's ready, remove the dutch oven from the smoker and let it rest for 20-30 minutes.
Using 2 forks or your hands with heat resistant gloves, shred the chuck roast and mix with the jus in the dutch oven.
Fill the mini french rolls with a handful of the shredded beef, top with a spoonful or more of Giardiniera and enjoy!
Notes
Use a top sirloin or top round if you cannot find a chuck roast.
Olive oil or hot sauce are great binder substitutes if you don't like mustard.
Use a roasting aluminum foil pan if you don't have a dutch oven. Make sure to seal it with a sheet of aluminum foil to enclose all the steam and allow the meat to tenderize.
Cook the braised meat on the stove top or in the oven if you don't want to continue using the grill.
The chuck roast is ready when it's fork tender. Take a fork, insert it into the chuck roast and give it a small twist. The meat should start to separate without any resistance. If it does not, close it again and continue cooking. Check again after 30 minutes.