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Place the mushrooms in a small bowl, pour the warm water over them, and let sit until softened, at least 30 minutes. Transfer the mushrooms to a cutting board, reserving the soaking liquid. Discard the mushroom stems and chop the caps.
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Put the spinach in a large heatproof bowl, pour boiling water over the spinach, and let the leaves wilt for 30 seconds. Drain, rinse with cold water, drain again, then squeeze out any excess moisture with your hands.
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In a small bowl, combine 1/4 cup of the reserved mushroom soaking liquid, the salt, white pepper, sugar, soy sauce, and sesame oil. Stir to dissolve the sugar. Set the sauce aside.
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In a wok or large skillet over medium heat, warm the canola oil. Add the ginger and stir-fry until aromatic, about 30 seconds. Add the mushrooms, spinach, carrot, and tofu. Stir to combine and then add the sauce. Stir-fry until a small amount of liquid remains, about 2 minutes. Stir the cornstarch mixture, then add it to the wok. When the mixture thickens, turn off the heat and add the green onions. Cool completely before use.