1: Set up your grill or smoker for indirect grilling and heat to 350 degrees.
2: Using a sharp knife, score the duck skin by running your knife in parallel lines, each about 1/2 inch apart. Turn the duck breast at a 90-degree angle and score once more. (It’s like scoring the skin on a ham.) Do not nick the flesh. Arrange on a rimmed sheet pan and brush on both sides with sesame oil.
3: Make the scallion brushes: Add cold water and ice to a medium mixing bowl. Trim the root end, then cut a 3-inch length off the bottom of each scallion. (Reserve the greens.) Make a series of 1-inch cuts through the white of each scallion, rotating it to form the “bristles.” Repeat on the opposite (green) end. Place in the ice water and refrigerate for 1 to 2 hours, or until the bristles separate and curl. Thinly slice the reserved greens crosswise to use as a garnish.
4: Make the Asian 5-4-3-2-1 rub: Combine the ingredients in a small mixing bowl and mix with your fingers or a wooden spoon. (Be sure to break up any lumps of sugar.)
5: Make the smoking mixture: In a medium mixing bowl, combine the wood chips (unsoaked), rice, tea, cinnamon sticks, cardamom pods, and black pepper, and set aside until ready to cook.
6: Season the duck leg quarters on both sides with the rub.
7: When ready to cook, brush and oil the grill grate well. Add the smoking mixture to the coals. Arrange the duck leg quarters on the grill grate away from direct flames. Smoke-roast the duck leg quarters for 1 1/2 to 2 hours, or until the meat reaches 185 degrees and the skin is crisp.
8: While the duck cooks, make the Hoisin Barbecue Sauce: Combine the ingredients in a medium saucepan and stir to combine over medium-high heat. Bring to a boil, then reduce the heat to a simmer. Simmer the sauce for about 5 minutes, or until it is concentrated and rich-tasting.
9: Transfer the duck leg quarters to a platter or plates. Using the scallion brushes, paint each leg with the Hoisin Barbecue Sauce. Sprinkle with the scallion greens. Serve with one or two bok choy halves.
