The seductively savory combination of soy sauce, sake, butter, and garlic seasons this chicken-and-egg fried rice. It's a complete meal that the whole family will devour. Day-old rice works best in this recipe, so plan ahead.
In a small bowl combine soy sauce and sake. In a separate bowl combine butter and minced garlic. Set both aside.
Heat a large cast iron pan or wok over high until very hot, about 2 minutes (a sprinkle of water should evaporate immediately). Add 2 teaspoons oil and swirl to coat. Beat together eggs and 1 teaspoon soy mixture. Add egg mixture; cook until set, about 1 to 2 minutes. Transfer cooked eggs to a cutting board (reserve skillet); roll up, and thinly slice eggs crosswise. Set aside.
Increase heat to high, add chicken in one layer; season with salt. Add half the butter mixture. Cook turning once until browned on both sides, 1 1/2 to 2 minutes. Add 1 tablespoon soy mixture; stir to coat. Remove to board with eggs. Add onion, carrot, scallion, and remaining butter mixture; cook 2 minutes until just beginning to brown. Add rice, cook, tossing occasionally and scraping rice from bottom of pan, until heated through, about 3 minutes. Add back chicken and scrambled egg and remaining soy sauce mixture. Toss to coat and heat through. Stir in scallion greens. Season with salt and pepper. Garnish with additional scallion greens.