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1
Bring water to a simmer in a small saucepan, then stir in mushrooms and simmer 10 minutes. Cool to room temperature, then strain through a sieve into a large bowl. Discard stems, then chill mushroom caps, covered, until ready to braise chicken. Stir remaining marinade ingredients into mushroom liquid, then add chicken and marinate, chilled, stirring occasionally, 1 day.
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2
Remove chicken from marinade and pat dry (discard marinade).
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3
Heat oil in a 12-inch heavy nonstick skillet over medium-high heat until hot. Just before browning each batch of chicken, sprinkle with 1 teaspoon kosher salt. Brown, skin side down first, about 3 minutes per side. Transfer to a plate.
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4
Pour off all but 1 tablespoon oil from skillet, then reduce heat to medium-low and saute ginger, stirring, 1 minute (do not brown). Add white wine and Shaoxing and boil until reduced by three fourths, about 5 minutes. Add broth, 1 1/2 cups water, sugar, soy sauce, white pepper, and remaining 2 teaspoons kosher salt and bring to a simmer.
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5
Combine cornstarch and remaining 2 tablespoons cold water and whisk into sauce, then boil 1 minute.
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6
Transfer chicken, skin side up, to a 6-to 8-quart heavy pot (place white meat on top). Add sauce, chestnuts, and reserved mushroom caps. Gently simmer, covered, until chicken is just cooked through, 15 to 25 minutes.
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7
Cut breasts in half and return to pot. Add ham. Gently simmer 3 minutes more.
