Combine sauce ingredients in a medium bowl; set aside.
Blanch broccoli in boiling water until tender-crisp, about 1 minute. Drain, rinse with cold water and drain again. Set aside.
Place a wok or stir-fry pan over medium-high heat until hot. Add oil, swirling to coat sides. Add ginger and onion salt; cook, stirring, until fragrant, about 10 seconds. Add all mushrooms; stir fry until mushrooms are tender, about 3 minutes.
Carefully add tofu and sauce; reduce heat and cook until heated through. Add cornstarch solution if sauce needs to be thickened.
Arrange broccoli in center of a serving platter. Surround with mushroom-tofu mixture.