Start by brining the eye of round using a wet brine. Mix water, salt and sugar in a pan and bring it to a boil so the sugar and salt dissolves completely. Leave it to cool off completely (important for food safety), then place the meat in the pan and store it in the fridge overnight. This is a 10% brine, which means a little saltier than usual (6%). That is on purpose.
Light up your grill and prepare it for indirect heat. I’m using the snake method with a brick here as divider in a Weber 18.5 inch grill.
Make sure your cooking temp is roughly 230-260° F. Place the meat on the grate, smoke it with some wood chips or chunks for roughly 2 hours, or until the inner temperature is 130° F, no higher.
When done, wrap it in aluminum foil and let it rest for 20 minutes.
Slice the round in thin slices and enjoy on a sandwich, or with boiled potatoes.