Soak cedar plank in water for at least 2 hours, or up to 24 hours, prior to grilling.
Stir the brown sugar and Dijon mustard together until smooth.
Turn grill to medium-high heat. When ready to cook, place the pre-soaked cedar plank on the hot grate and close the lid. Leave it there until there is a smoky smell, about 4-5 minutes.
Turn the plank over and place the salmon, skin side down, on top. Spread a spoonful of mustard mixture on top of each filet and add a thin slice of lemon, if desired.
For indirect heat grilling: Heat grill to medium-high heat. Cook salmon in the center of hot grate, away from the heat. Cover the grill and cook salmon for about 15 to 25 minutes (depending on thickness), or until cooked through (135 degrees F) and flakes easily with a fork. Be careful not to overcook it.
You can cook the salmon over direct heat, but be careful to check the cedar plank occasionally. If the edges start to catch fire, mist with water and move the plank to a cooler part of the grill.
*You can place as many salmon fillets on the cedar plank as it will fit. Cedar planks come in different sizes.To cook in the oven: Prepare the cedar planks and salmon as directed but instead of using a grill, preheat oven to 375 degrees and cook the salmon for about 15-25 minutes (depending on thickness) or until it easily flakes with a fork.