Heat 1 tablespoon of vegetable oil in a wok or large skillet over high heat. Stir in the mushrooms, snow peas, water chestnuts, and bok choy; season to taste with salt and pepper. Cook and stir until the vegetables are just tender, about 5 minutes. Remove the vegetables from the wok and wipe the wok clean.
Heat the remaining 1 tablespoon of vegetable oil in the wok. Stir in the garlic and ginger; cook a few seconds until the garlic begins to turn golden brown. Stir in the chicken and cook until the chicken is no longer pink, about 5 minutes. Add the wine, sugar, and chicken broth; bring to a boil. Dissolve the cornstarch in the water and stir into the simmering sauce. Once the sauce returns to a simmer, stir until thick and clear, about 30 seconds. Return the vegetables to the wok and toss until hot and coated with the sauce.