Melt the butter in a saucepan over medium heat, and mix in the brown sugar, lemon juice, dill, and cayenne pepper. Cook and stir until the brown sugar is dissolved. Reduce heat and simmer until ready to brush onto the salmon.
Season salmon with lemon pepper, and place directly on the grill grate, skin side down. Brushing frequently with the butter and brown sugar glaze, cook the salmon 20 minutes, or until easily flaked with a fork. Reserve some of the glaze for serving.