Asian Cabbage and Shiitake Stir-Fry

1 1/2 tablespoons soy sauce

1 tablespoon unseasoned rice vinegar

1 1/2 teaspoon cornstarch

1 teaspoon sugar

2 tablespoons vegetable oil

1 large clove garlic, finely chopped

2 teaspoons finely chopped peeled ginger 3- to 4-ounce fresh shiitake mushrooms, stems discarded and caps cut into 1/4-inch wide strips

2 pounds Napa cabbage, quartered lengthwise and cored, then cut crosswise into 1-inch wide pieces, about 15 cups 4 scallions, thinly sliced

1/2 teaspoon Asian sesame oil, optional