Heat a large nonstick skillet or wok over high heat and then swirl in 1 tablespoon of the oil. When the oil begins to smoke, add the egg and swirl the skillet to make a flat egg pancake that's only half set, about 10 seconds; scoop onto a plate.
Add the tofu to the skillet, season with 1/4 teaspoon salt and stir-fry until golden, about 4 minutes. Transfer to a bowl. Add the remaining 1 tablespoon oil to the skillet, then add the shiitakes and cook until golden, about 3 minutes. Add the ginger, garlic, soy sauce and 1/4 teaspoon salt and stir-fry until fragrant, about 1 minute.
Add the rice, stirring to break up any clumps, and spread it out in the skillet. Cook until heated through, about 3 minutes. Return the egg, tofu and any juices to the skillet and cook, stirring to break up the egg and mix it in thoroughly, about 1 minute. Season with additional salt if needed. Sprinkle with the scallions and serve.
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