Preparation Stir together all sauce ingredients until brown sugar is dissolved, then set aside. Toast sesame seeds in a dry 12-inch heavy skillet (not nonstick) over medium heat, stirring, until pale golden, then transfer to a small bowl. Heat oil in skillet over medium-high heat until it shimmers, then sauté ginger and garlic, stirring, until fragrant, about 30 seconds. Add shiitakes and sauté, stirring frequently, until tender and starting to brown, about 6 minutes. Reduce heat to medium, then add cabbage and most of scallions (reserve about a tablespoon for garnish) and cook, stirring occasionally, until cabbage is crisp-tender, about 6 minutes. Add sauce and simmer 2 minutes. While cabbage is cooking, cook soba and edamame together in a pasta pot of boiling salted water (2 tablespoons salt for 6 quarts water) until noodles are just tender, about 6 minutes. Drain in a colander and rinse under cool water to stop cooking and remove excess starch, then drain well again. Transfer to a large bowl and toss with sesame seeds and vegetable mixture. Serve sprinkled with reserved scallions. Cooks& note:If you aren't able to find Korean hot-pepper paste, substitute 3/4 teaspoon Chinese chile paste and reduce the amount of soy sauce to 1/4 cup.