Soak the wood chips in water for at least 30 minutes.
Fill a medium saucepan with salted water and bring to a boil. Rinse the farro in a fine-mesh sieve under cold water, add it to the saucepan, and boil until fully cooked but still al dente, according to package directions. Drain in a fine-mesh sieve and let cool.
In a small bowl combine 1½ teaspoons salt, ½ teaspoon pepper, the ginger, paprika, and cardamom. Slip your fingers beneath the skin and fat on the duck breast to loosen it from the meat. Using kitchen scissors, trim away some of the skin and fat, leaving a 1- to 1½-inch-wide strip of skin and fat that runs the entire length of each duck breast. Make four to five diagonal slashes in the skin and fat on each breast, slashing down to but not through the meat. This trimming and slashing procedure will help prevent flare-ups and allow the fat to melt and the skin to crisp during grilling. Lightly brush the duck breasts with oil and season evenly with the spices.
Prepare the grill for direct and indirect cooking medium heat (350° to 450°F).
Brush the cooking grates clean. Drain and add the wood chips to the charcoal or to the smoker box of a gas grill, following manufacturer’s instructions, and close the lid. When the wood begins to smoke, cook the duck breasts, skin side down first, over direct medium heat, with the lid closed, until the skin is browned and the meat is cooked to your desired doneness, 8 to 10 minutes for medium rare, turning once (if flare-ups occur, move the breasts temporarily over indirect heat). Remove from the grill and let the duck rest while you finish the salad.
In a large skillet over medium-high heat, warm 2 tablespoons oil. Add the scallions, bell pepper, and red pepper flakes and sauté until the vegetables are softened, 2 to 3 minutes. Add the cooled farro and the vinegar. Season with salt and pepper. Sauté for 1 to 2 minutes more. Remove the skillet from the heat.
Cut the duck breasts into slices and serve warm with the farro and vegetable salad.
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