Make dressing: Whisk together all dressing ingredients in a large bowl and let stand at room temperature while making salad.
Cook rice: Rinse rice well in a large sieve under cold water, then drain. Heat butter in a 4- to 5-quart heavy pot over moderate heat until foam subsides, then cook onion, stirring occasionally, until golden, about 5 minutes. Add rice and cook, stirring, until fragrant, about 3 minutes. Stir in water and broth and bring to a boil, then reduce heat and simmer, covered, until rice is tender, 1 to 1 1/4 hours. (Grains will split open but not all liquid will be absorbed.) Drain well in a colander and cool to warm before adding to dressing. (Spread rice out in a shallow baking pan to cool faster.)
Cook sugar snaps and duck while rice is cooking: Preheat oven to 375°F.
Cook sugar snaps in a 4-quart pot of boiling salted water until crisp-tender, about 2 minutes. Drain in a sieve and rinse under cold water to stop cooking, then drain well.
Pat duck dry and season with salt and pepper. Score skin in a crosshatch pattern and place duck, skin sides up, in a lightly oiled shallow (1-inch-deep) baking pan. Roast in middle of oven until an instant-read thermometer inserted horizontally into center registers 120°F (for medium-rare), about 25 minutes. Leave oven on.
Transfer duck to a cutting board and when just cool enough to handle, remove skin. Thinly slice skin (scored side down), keeping duck breast warm, loosely covered with foil. Bake skin in baking pan in middle of oven until very crisp, about 15 to 20 minutes, then transfer with a slotted spoon to paper towels to drain.
Cut duck breast halves in half horizontally (butterfly-style), then cut across the grain into thin slices. Add duck and any juices to dressing along with rice, sugar snaps, scallions, apricots, pecans, and salt and toss gently to combine.
Just before serving, scatter crisp duck skin on top.
* Available at D'Artagnan (800-327-8246) and Hudson Valley Foie Gras (877-289-3643).