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Whisk together stock and cornstarch in a small bowl until smooth.
Heat 1 tablespoon canola oil in a large cast-iron skillet over high. Add steak; cook, stirring occasionally, until browned, 2 to 3 minutes. Add 1 1/2 tablespoons soy sauce and 1 teaspoon sugar. Cook, stirring often, until meat is charred, 1 to 2 minutes. Remove steak to a plate. Add cabbage, ginger, garlic, and remaining 1 tablespoon canola oil; cook, stirring often, until cabbage begins to wilt, 2 to 3 minutes. Add stock mixture, remaining 2 tablespoons soy sauce, and remaining 1/2 teaspoon sugar. Bring to a boil; reduce heat to medium, and simmer until cabbage is crisp-tender, about 2 minutes. Stir in steak, carrots, scallions, sesame oil, and vinegar. Cook 1 minute. Serve over brown rice. Sprinkle with sesame seeds.