Whisk together all dressing ingredients in a large bowl and let stand at room temperature while making salad.
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. Add onion and stir occasionally until golden, about 5 minutes. Add rice and stir 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.)
While rice is cooking, prepare sugar snap peas 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.
Slice the duck cut across the grain into thin slices. Add duck and any juices to dressing along with rice, sugar snap peas, scallions, apricots, pecans, and salt; toss gently to combine.
Just before serving, scatter crisp duck skin on top.