In a 5- to 6-quart pan, bring about 2 quarts water to a boil over high heat; add potatoes. Cover and simmer over low heat until potatoes are tender when pierced, 15 to 20 minutes. Drain and immerse in cold water. When potatoes are cool, drain well. Use or, if making ahead, cover and chill up to 1 day.
Soak the onions about 15 minutes in cold water to cover. Drain and mix onions with rice vinegar. Cut potatoes in quarters; add to onions.
Trim tender watercress sprigs from stems, then finely chop enough of the coarse stems to make 1/2 cup (discard extras or save for other uses). Mix chopped stems with potato salad. Mound watercress sprigs on a large oval platter with potato salad alongside; cover and keep cool.
Rinse salmon and pat dry. Place, skin side down, on a piece of heavy foil. Cut foil to follow outlines of fish, leaving a 1-inch border. Crimp edges of foil to fit up against edge of fish. Mix soy sauce with brown sugar and brush onto the salmon fillet.
On a charcoal barbecue. Mound and ignite 50 to 60 charcoal briquets on the firegrate of a barbecue with lid. When briquets are spotted with gray ash, about 20 minutes, bank 1/2 of the coals on each side of the firegrate. Drain wood chips and sprinkle equally onto mounds of coals. Place grill 4 to 6 inches above coals.
On a gas barbecue. Turn gas heat to high. Place wood chips in the barbecue's metal smoking box or in a small shallow foil pan, and set directly on the heat in a corner. Close lid until barbecue is hot, about 10 minutes. Adjust gas for indirect cooking (no heat down center).
Lay fish on center of grill, not over coals or flame. Cover barbecue (open vents for charcoal) and cook until fish is barely opaque in thickest part (cut to test), 15 to 20 minutes. Transfer fish to a platter with salad. Add salt to taste. Serve hot or cool.