Wipe meat with a damp towel, then cut crosswise into 2 or 3 equal pieces.
Prepare barbecue for direct heat.
If using charcoal briquets, cover firegrate with a single, solid layer of ignited coals; let them burn down to desired heat. Set grill in place and measure heat.
If using a gas barbecue, turn all burners to high and close lid for 10 minutes. Adjust burners to desired heat. Set grill in place and measure heat.
When grill is medium-hot (you can hold your hand at grill level only 3 to 4 seconds), lay steak on barbecue. Cook, turning as needed to brown evenly, until meat is pink in center of thickest part (cut to test), 7 to 10 minutes.
Meanwhile, in a shallow pan (at least 9 in. wide) on grill, combine butter, mustard, vermouth, and Worcestershire; stir occasionally until butter melts, then push to a cool area of grill and keep warm.
Set cooked steak in pan, slice, and stir meat juices into sauce. Divide watercress among plates, lay meat on it, and spoon sauce over portions.