Make the Teriyaki glaze by placing all the glaze ingredients into a small saucepan.
Prepare the barbecue for direct cooking over high heat (250°C to 290°C).
Combine all the dressing ingredients together. Cover with plastic wrap and refrigerate until required.
Truss the beef into an even cylindrical shape with butchers string, if it has not already been done by the butcher. On a chopping board, sprinkle the salt, pepper, brown sugar and thyme leaves over the board. Roll the beef eye fillet on the chopping board, coating it all over in the seasonings.
Drain and add the wood chips to the charcoal or to the smoker box of a gas barbecue, following the manufacturer’s instructions. When smoke appears, barbecue the beef over direct high heat for 4 minutes, with the lid down, quarter turning the beef each minute or until just seared and caramelised.
Remove the beef from the barbecue. Using a very sharp chefs knife, thinly slice the beef. Lay each slice on the chopping board, press and flatten with the side of the knife to achieve very thin slices. Place the slices of beef on a platter plate. You can serve immediately or cover with plastic wrap and refrigerate until required, up to 6 hours.
Top with the rocket, parmesan shavings. Drizzle with the dressing and olive oil. Finish with a season of salt and pepper.