Stir together broth, soy sauce, cornstarch, and 1/2 teaspoon salt until cornstarch has dissolved.
Heat wok over high heat until a drop of water evaporates instantly. Pour peanut oil down side of wok, then swirl oil, tilting wok to coat side. Add garlic and stir-fry until pale golden, 5 to 10 seconds. Add half of bok choy and stir-fry until leaves wilt, about 2 minutes, then add remaining bok choy and stir-fry until all leaves are bright green and limp, 2 to 3 minutes total. Stir broth mixture, then pour into wok and stir-fry 15 seconds. Cover with lid and cook, stirring occasionally, until vegetables are crisp-tender, 2 to 4 minutes. Stir in sesame oil, then transfer to a serving dish.
Cooks' note: Baby bok choy can be washed, dried, and halved 1 day ahead. Chill, wrapped in paper towels, in a sealed bag.