Preheat a gas grill to high; adjust to medium low after 15 minutes. (If cooking over charcoal, allow the coals to burn until they are covered with white ash.) Place the salmon skin-side down on the cedar plank.
Combine the brown sugar, oil, thyme, and cayenne in a bowl. Spread over the salmon. Place the planked salmon on the grilling grate and cook, with grill covered, about 40 minutes or just until the surface fat begins to turn white.
Rainy-Day Method: Preheat oven to 325° F. Prepare the salmon as described above, then roast on the cedar plank for about 25 minutes.