Start by prepping all of your ingredients for the French dip and then trim your round of eye roast by removing all the silver skin and hard exterior fat, season with salt, pepper and garlic.
At this point, light some coals in a charcoal chimney and wait about 20 mins so that the coals are blazing hot and sear the round of eye on each side. Remove the round of eye and set your smoker up to 250 degrees.
While your smoker is setting up to 250 degrees place all of your French dip ingredients into an aluminum pan. This is going to be your drip pan as well.
Once the temp is 250 degrees place the drip pan under the racks where you are going to be placing your round of eye roast. You want to catch all of those juices from the steak while it's smoking and it's going to add an extra depth of flavor.
Smoke until an internal temperature of 128 degrees and then remove and place the round of eye roast on a resting rack and cover with aluminum foil, resting for a minimum of 15 mins. During this time make your horseradish sauce and remove all of the onion and carrots out of the pan and place in a separate bowl. Add the liquid juice into a bowl for dipping.
After resting, cut down the center and split the round of the eye roast in half. From here, slice thin slices to build your sandwiches with.
Now build your sandwich, Toasted buns, horseradish sauce, thin slices of steak (as much or little as you want), onions and carrots, Swiss cheese, horseradish sauce on the top bun too and place under the broiler until the cheese is melted.
Serve and enjoy with family and friends!
