Toss together chicken and salt in a large bowl; set aside. Heat oil in a wok or large cast iron skillet over medium. Add garlic and ginger; cook, stirring constantly, until fragrant, about 30 seconds. Push garlic and ginger to one side of wok; increase heat to high, and add chicken pieces in a single layer. Cook, undisturbed, until browned in spots, about 2 minutes. Continue cooking, stirring occasionally, until chicken is no longer pink, about 3 minutes.
Stir mushrooms, ⅔ cup water, Shaoxing wine, soy sauce, dark soy sauce, and sugar into wok. Bring to a boil over high. Reduce heat to medium; cover and simmer vigorously, undisturbed, until a thermometer inserted in the thickest piece of chicken registers 165°F, 5 to 8 minutes.
Uncover wok, and increase heat to high. Add peppers and sesame oil; cook, stirring occasionally, until liquid has reduced into a thick sauce, 3 to 5 minutes. Remove from heat, and stir in scallions. Season to taste with additional soy sauce. Serve with steamed rice on the side, if desired.
