Cut half of ginger into very fine matchsticks (less than 1/8 inch thick; about 1 tablespoon) and reserve. Grate remaining ginger and squeeze pulp with your fingers to yield 1 teaspoon liquid, then discard pulp.
Remove any bruised or withered outer leaves from bok choy. Trim 1/8 inch from bottom of each bok choy, then cut each head into quarters. Wash bok choy in several changes of cold water and dry in a colander or salad spinner until dry to the touch.
Whisk together ginger juice, chicken broth, rice wine, soy sauce, cornstarch, salt, and sugar in a small bowl until cornstarch is dissolved.
Heat wok over high heat until a bead of water vaporizes within 1 to 2 seconds of contact.
Pour oil down side of wok, then swirl oil, tilting wok to coat sides. Add ginger matchsticks and stir-fry 5 seconds.
Add bok choy and stir-fry until leaves are bright green and just limp, 1 to 2 minutes.
Stir broth mixture, then pour into wok and stir-fry until vegetables are crisp-tender and sauce is slightly thickened, about 1 minute. Remove from heat and drizzle with sesame oil, then stir to coat.