Soak a cedar plank in salted water for 1 to 2 hours, then drain. Rinse the salmon under cold running water and pat dry with paper towels. Generously season the salmon with salt and pepper. Lay the salmon skin-side down (if skin is still on) 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 to medium-high. Place the cedar plank in the center of a hot grate. Close the grill hood and cook for about 8 to 10 minutes. Check to be sure the cedar plank is only smoking and not catching fire or overly black around the edges. If so, turn the burner down or off under the plank and continue cooking (indirect cooking) with the other burner(s). Total cook time will be 15 to 25 minutes. The internal temperature should read 135 degrees F. With a spatula cut under the meat leaving the skin attached to the plank and transfer the salmon to a platter for serving.