If using charcoal briquets, ignite 40 briquets on the firegrate of a barbecue, let burn until coals are dotted with ash, about 20 minutes, then push half the coals to each side of the grate. Set a foil drip pan (about 8 by 12 in.--slightly longer and wider than duck--and at least 1 in. deep) on firegrate between mounds of coals. Set grill in place. Add 4 briquets to each mound of coals now and every 30 minutes of cooking.
If using a gas barbecue, turn heat to high and close lid for at least 10 minutes. Adjust heat to medium and burners for indirect cooking. Put drip pan between ignited burners. Set grill in place.
Meanwhile, remove giblets from 1 duck (about 5 lb.) and reserve for other uses. Rinse, drain, and pat bird dry. Discard lumps of fat. With a fork, pierce just through skin all over at about 1-inch intervals. Fold wing tips under back. Pin neck skin to back of duck with a metal skewer. Pin body cavity shut and tie legs together.
Lay duck, breast up, on grill directly over drip pan. Cover barbecue; open vents for charcoal. Temperature inside barbecue should be about 300°. Cook until duck is lightly browned and a thermometer inserted in thigh against bone at hip joint registers 185°, about 2 hours.
Transfer duck to a platter. Let cool at least 2 hours. If making ahead, cover and chill up to 1 day. Discard drippings.
Set duck, breast up, in a V-shaped rack in a shallow pan about 8 by 12 inches.
Bake duck in a 350° oven until very richly browned, about 55 minutes.
Transfer to a platter. Slice skin off in large pieces, trimming fat off the underside. Cut the meat from the duck and season with salt and pepper to taste; accompany meat with hoisin or plum sauce.