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. Brush over the salmon.
Place the planked salmon on the grilling grate and cook, with the grill covered, until mostly cooked through but still slightly pink in the middle, about 40 minutes.