The day before you plan on making the sandwiches, place the meat in a large glass container. Add all of the marinade ingredients to the container with the meat.
Use your hands to rub the marinade onto the meat, coating all the sides. Cover the container with an airtight lid and let the meat marinade in the fridge overnight.
When it's time to grill the meat, set the container that contains the meat out on the counter while your grill preheats. Preheat the grill over medium heat to about 375-400°F.
Once the grill is preheated, spray the grill grates with avocado oil cooking spray. Place the meat directly on the grill grates and close the grill.
Wait 5-7 minutes, then flip the meat and baste the meat with some of the leftover marinade mixture. In 5-7 more minutes, check the internal temperature of the meat with a digital meat thermometer.
If it has not reached your desired temperature for doneness (135°F for medium rare), the continue the flipping and marinating process until the meat reaches your desired internal temperature. Wait about 3-4 minutes between each time you check the meat.
Once the meat is done, place it on a plate and cover it with aluminum foil.
While the meat rests, prepare the gluten-free sub rolls. Cut each one open and butter both sides if desired. Spread barbeque sauce onto one side of each roll. Break the cheese slices into smaller pieces and place them randomly on the non-barbequed half of each roll.
After the meat has rested for 15-20 minutes, uncover it and transfer it to a cutting board. Use a sharp knife to shave off thin slices of the meat. Once the meat has been shaved into thin slices, start making the sandwiches!
Fold and pile the thin slices of meat onto one half of each roll (doesn't matter if it's the cheese or bbq half). Salt the meat. Sandwich each roll together and enjoy!
Note: If you want to toast your steak sandwich, I recommend wrapping the sandwich in aluminum foil to keep the sandwich together while it toasts. Alternatively, you can toast the two halve of each sandwich separately and then sandwich them together after toasting.