In a 10- to 12-inch frying pan, combine onion, vermouth, mustard, tarragon, and sugar. Stirring often, boil over high heat until reduced to 3/4 cup, 5 to 8 minutes.
Rinse salmon and pat dry. Lay fish open, skin down, on a sheet of foil (about 12 by 17 in.). Trim foil to fit salmon. Spread onion mixture over top of salmon.
If using charcoal briquets, mound and ignite 60 briquets on the firegrate of a barbecue with a lid. When briquets are dotted with gray ash, in 15 to 20 minutes, push equal amounts to opposite sides of firegrate. Add 5 more briquets to each mound of coals now and every 30 minutes while cooking. Set a drip pan on firegrate between coals. If using a gas barbecue, turn all burners to high and close lid for 10 minutes. Adjust burners for indirect cooking (no heat down center) and keep on high unless recipe specifies otherwise. Set barbecue grill in place. Set foil with salmon on grill but not over heat.
Cover grill (open vents for charcoal). Cook until fish is opaque but moist-looking in thickest part (cut to test), 30 to 40 minutes.
Slide a rimless baking sheet under salmon and foil, then slide fish onto a large flat platter or board. Cut salmon and lift portions from skin with a wide spatula. Season to taste with salt.