Trim bases of bok choy and separate outer leaves from stalks, leaving the smallest inner leaves attached. Rinse and thoroughly dry bok choy in a salad spinner.
Heat a wok or large frying pan (not nonstick) over medium-high heat until hot when you wave your hand over the bottom. Add vegetable oil, garlic, and ginger and stir once; then immediately add mushrooms and stir-fry until they just begin to brown, 1 to 2 minutes.
Add rice wine and cook 30 seconds. Add bok choy leaves and stalks and cook, tossing with tongs, until beginning to wilt, about 1 minute. The wok may seem crowded, but the leaves wilt quite a bit.
Add soy sauce, sesame oil, salt, and pepper; cook, tossing often, until bok choy is tender-crisp, another 1 to 1 1/2 minutes.