Braised Chicken with Smoked Ham, Chestnuts, and Ginger

12 dried Chinese black mushroom s

1/2 teaspoon white pepper

2 teaspoons kosher salt , divided

1 tablespoon soy sauce

2 teaspoon sugar

1 cup sliced peeled ginger

1/4 cup Shaoxing wine or dry Oloroso Sherry

1/4 cup dry white wine

2 small chicken s (2 1/2 to 3 pounds), each cut into 6 pieces (legs, thighs, and breasts without backbones) For cooking chicken and sauce:

2 3/4 cups water

2 tablespoons corn oil

1 tablespoon plus 2 teaspoon kosher salt , divided

1/4 cup finely julienned peeled ginger

1/2 cup dry white wine

3/4 cup Shaoxing wine or dry Oloroso Sherry

2 1/2 cups reduced-sodium chicken broth

1 1/2 cups plus 2 tablespoon cold water , divided

3 1/2 teaspoons sugar

2 tablespoons soy sauce

1 teaspoon white pepper

2 tablespoons cornstarch

7 to 8 ounces bottled roasted chestnuts

1/4 pound thinly sliced country ham or speck

Accompaniment: steamed jasmine rice