First thing in the morning, set out some butter to come to room temperature.
Start your smoke tube (I like pecan wood in mine) and let it get going well.
Combine all the dry ingredients for the roast beef and sprinkle it evenly all over the eye of round roast. Place in your grill with the smoke tube for an hour prior to preheating your grill.
Insert your temperature probe into the thickest part of the beef and heat your grill to 225 degrees. Cook until the internal temperature of the beef hits 145 degrees.
If you haven't already, render your fat until you have about 1/4 cup of liquid. Discard the solids.
Once your meat has reached temperature, remove it from your pit and cover loosely with aluminum foil. Let rest for 20 to 30 minutes.
As your meat rests, make your jus. Heat the rendered beef-fat over medium-high heat until very hot. Add the flour and whisk until it is fully incorporated and begins to thicken. Whisk in the beef broth, salt, and pepper. Whisk until it begins to thicken, then turn the heat down to very low so that it stays hot for serving.
Slice the meat in your deli slicer on the thinnest setting that still allows you to have fully intact slices.
Preheat your broiler on high. Butter all of the buns and sprinkle lightly with kosher salt. Place on a sheet pan butter side up and toast under the broiler, being careful not to burn them.
Take your buns out of the oven and set the tops aside. Pile the bottoms with sliced meat and two slices of provolone. Place in the middle rack under the broiler long enough for the cheese to start melting, then move to the top rack and lightly toast it.
Remove from the oven and place the buns tops on each sandwich. Serve with the jus in wide bowls on the side for dipping.