How to Make It
Pat duck dry, and place on a cutting board, breast side down. Using kitchen shears, cut down each side of the backbone. Remove and discard backbone (or reserve for stock). Open duck up to expose breastbone; trim and discard excess fat. Using a sharp knife, cut lengthwise through center of breastbone. Continue cutting lengthwise to separate duck into 2 halves. Using your fingers, loosen skin from duck, being careful not to tear skin. Using a metal or wooden skewer, prick skin all over at a 45-degree angle, making sure not to pierce meat. Rub both sides of duck halves with jerk seasoning, and place, skin sides up, in a large pan. Chill, uncovered, 8 hours or overnight.
Let duck stand at room temperature while preheating charcoal grill, about 30 minutes. Open vents of grill completely. Fill an 8-inch-square disposable aluminum pan with 1 inch of water. Place water pan on one side of bottom grill grate. Light charcoal chimney starter filled halfway with briquettes. When briquettes are covered with gray ash, pour onto empty side of bottom grate. Place a few pecan or cherry wood chunks on top of hot coals; insert top grate. Cover grill, and adjust bottom and top vents to bring internal temperature to 225°F to 250°F. Place duck halves, skin sides up, on grate over water pan. Cover and smoke, maintaining an internal grill temperature of 225°F to 250°F, 30 minutes.
Line 2 sheets of aluminum foil with parchment paper. Transfer each duck half to a prepared foil sheet; wrap foil sheets tightly to seal and create packets. Place duck packets on grate over water pan, and continue smoking, covered, at a grill temperature of 225°F to 250°F, until a meat thermometer inserted in thighs and thickest portions of breast registers 135°F, about 45 minutes to 1 hour and 10 minutes. Transfer duck packets to a small insulated bag or cooler, and let rest 30 minutes.
While duck rests, light a charcoal chimney starter filled halfway with briquettes. When briquettes are covered with gray ash, remove top grate of grill, and pour briquettes over remaining coals on bottom grate. Replace top grill grate. Open bottom vent completely to increase internal grill temperature to medium (350°F to 400°F) with lid removed.
Unwrap duck, and discard foil packets. Place duck halves, skin sides down, over side of grill with coals. Grill, uncovered, turning often, until skin is browned and crispy and a meat thermometer inserted in thighs and thickest parts of breast registers 155°F, about 10 minutes. Remove duck halves from grill, and let rest 10 minutes.
Season cut sides of peach halves with 1/4 teaspoon salt. Place peaches, cut sides down, on oiled grill grates over side with coals, and grill, uncovered, until charred and softened, 4 to 8 minutes. Transfer 2 grilled peach halves to a blender; add mustard, honey, brown sugar, vinegar, and remaining 1/2 teaspoon salt. Process until smooth, about 20 seconds. Serve duck with peach barbecue sauce and remaining grilled peach halves.