Heat a large (12-inch) nonstick skillet or wok over high heat. Add 1-1/2 teaspoons of the oil and heat until very hot. Add half of the beef. Cook, stirring often, until seared on all sides, about 3 minutes. Transfer to a plate and set aside. Repeat with another 1-1/2 teaspoons oil and the remaining beef.
Add the remaining 1 tbsp oil to the skillet and heat. Add the onion and mushrooms and stir-fry until the onion softens, about 3 minutes.
Stir in the beef broth, soy sauce, sake, and sugar and bring to a boil. Add the spinach, tofu, and bean sprouts and cook until the spinach is wilted, about 1 minute. Stir the cornstarch mixture, add to the skillet, and cook until the sauce is slightly thickened. Season with the pepper.
Spoon the rice into individual dinner plates, top with the beef-vegetable mixture, and serve immediately.