Because the heat on individual grills varies, cooking times will range from about 30 minutes to one hour. Be sure to regulate the barbecue so that the wood chips smoke and the briquettes burn but do not flame. When the salmon is ready, it will have a nice firm texture and a lovely sweet glaze.
Start one day ahead because the salmon needs to sit in brine overnight.
Stir first 4 ingredients in large bowl until both sugars dissolve. Add salmon, skin side up, to brine, pressing to submerge. Cover and refrigerate overnight.
Remove salmon from brine; discard brine. Rinse salmon under cold water.
Place salmon, skin side down, on rack. Let stand until top is dry to touch (do not pat dry), about 1 hour.
Prepare barbecue (medium-low heat). Using 2 layers of foil, make 12 x 10-inch foil rectangle with 1-inch-high sides. Place 3 cups smoke chips on foil rectangle. Set atop coals 5 minutes before grilling salmon. Position barbecue rack at least 6 inches above briquettes. Position vents on barbecue so that chips smoke and briquettes burn but do not flame.
Using 3 layers of foil, make another 12 x 10-inch foil rectangle with 1-inch-high sides. Using skewer, pierce 6 holes evenly over bottom of rectangle. Arrange salmon, skin side down, on rectangle. Place salmon on foil on grill. Cover; cook until salmon is firm to touch and glaze forms over salmon, adding more smoke chips to barbecue if necessary, 30 minutes to 1 hour, depending on heat of grill.
Remove salmon from foil, leaving skin on foil. Transfer salmon to platter; serve warm or at room temperature. (Can be made 1 day ahead. Cover; chill. Bring to room temperature before serving.)