Marinate sirloin in tamari with 1 teaspoon ginger, the garlic, 1 scallion, and the red-pepper flakes. Meanwhile, bring a large saucepan of water to a boil. Cook remaining scallions with the carrots and cabbage until crisp-tender, about 2 minutes. Transfer to a plate using a wire-mesh skimmer.
Heat a medium-heavy skillet over high heat; add 1 teaspoon oil, and sear sirloin, 1 to 1 1/2 minutes per side. Transfer to plate. Return skillet to medium heat, and add remaining 1/2 teaspoon oil and 2 teaspoons ginger. Cook until sizzling. Stir in rice and 1/2 teaspoon salt. Stir in scallion-sirloin mixture and any juices. Add water, and cook, covered, until rice is heated through and sizzling, 7 to 8 minutes.
Uncover, and make 4 indentations in the rice with the back of a spoon. Crack 1 egg into each, and cook, covered, until whites are set but yolks are still runny, about 4 minutes. Stir eggs into rice.