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 is only half set, about 10 seconds; scoop onto a plate.
Add the beef to the skillet, season with 1/4 teaspoon salt and stir-fry, breaking up any clumps, until brown, about 4 minutes. Transfer to a bowl. Add the remaining tablespoon oil, then add the broccoli, garlic, ginger, soy sauce and 1/4 teaspoon salt and stir-fry until the broccoli is crisp-tender, 2 to 3 minutes.
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 and the beef and any juices to the skillet and cook, stirring and breaking up the egg to mix it in thoroughly, about 1 minute. Season with additional salt if needed. Sprinkle with the scallions and peanuts and serve.