Toss the chicken with the oyster sauce in a bowl. Mix the cornstarch with 3 tablespoons cold water in another bowl. Place both bowls near the stove with the remaining ingredients.
Heat a wok or deep skillet over high heat until hot. Add the peanut oil, then the scallions, ginger and garlic, and stir-fry about 20 seconds. Add the chicken mixture and stir-fry until the meat is no longer pink on the outside, 2 to 3 minutes. Stir in the mushrooms and bok choy.
Add the broth and sesame oil to the skillet and bring to a boil over high heat. Add the cornstarch mixture, return to a boil and cook, tossing, until the chicken is cooked through and the sauce is thick and glossy, 2 to 3 minutes. Serve with rice, if desired.