Soak cedar plank in salted water for 2 hours, then drain. Remove skin from salmon fillet. Remove any remaining bones. Rinse the salmon under cold running water and pat dry with paper towels. Generously season the salmon with salt and pepper on both sides. Lay the salmon (on what was skin-side down) on the cedar plank and carefully spread the mustard over the top and sides. Place the brown sugar in a bowl and crumble between your fingers, then sprinkle over the mustard.
Set grill for indirect grilling and heat to medium-high. Place the cedar plank in the center of the hot grate, away from the heat. Cover the grill and cook until cooked through, around 20 to 30 minutes. The internal temperature should read 135 degrees F. Transfer the salmon and plank to a platter and serve right off the plank.
Cook's Note: A direct method to grill the salmon may be used. Soak the cedar plank well. Spread the mustard and brown sugar on the salmon, but do not place the fish on the plank. Set up the grill for direct grilling on medium-high. When ready to cook, place the plank on the hot grate and leave it until there is a smell of smoke, about 3 to 4 minutes. Turn the plank over and place the fish on top. Cover the grill and cook until the fish is cooked through, reaching an internal temperature of 135 degrees F. Check the plank occasionally. If the edges start to catch fire, mist with water, or move the plank to a cooler part of the grill.