Set the grill for indirect heat, preheating to very hot (500 degrees), then turn it down to 300 degrees, or as low as you can go.
Prepare one large foil tray for cooking the salmon. [Make your own: Layer two 16-inch pieces of foil (one directly on top of the other) and fold up each side about an inch, to create a rectangular “tray.”
Pat-dry the salmon (with paper towel). Place on the foil tray.
In a small bowl mix together the butter, shallots, lime, dill, brown sugar, and pepper. Press evenly over the salmon. He...