In a medium bowl, whisk together 3 tablespoons of the sake and 1 teaspoon soy sauce. Add the chicken and toss. Let stand at room temperature for 20 minutes or cover and refrigerate for up to 2 hours.

Meanwhile, in a small saucepan over medium, combine the remaining 1 tablespoon sake, 3 tablespoons of soy sauce, the mirin and sugar. Cook, stirring, until the sugar is dissolved, about 1 minute. Off heat, stir in the ginger; set aside.

In a medium bowl, toss the cabbage and scallions with the remaining 1 tablespoon soy sauce, the rice vinegar and sesame oil. Set aside. Drain the chicken in a fine mesh strainer. Wipe out the bowl, then return the chicken to it. Sprinkle with cornstarch and toss to coat.

In a 12-inch nonstick skillet over medium-high, heat 2 teaspoons of the oil until beginning to smoke. Add half the chicken in an even layer and cook without stirring until well browned on the bottom and the edges turn opaque, 3 to 4 minutes.

Flip and cook without stirring until well browned on the second side, about another 3 minutes. Transfer to a clean bowl and repeat with the remaining 2 teaspoons oil and remaining chicken.

Wipe out the skillet, then return the chicken to the pan. Pour in the soy sauce-ginger mixture and stir to coat. Cook over medium-high, stirring, until the liquid is syrupy and the chicken is glazed, about 2 minutes.

Remove from heat. Divide the rice among 4 bowls. Top with the cabbage mixture and chicken.