Chicken Yakitori with Cold Soba Noodles

Name: Chicken Yakitori with Cold Soba Noodles
Ingredients: 1/3 cup plus 2 tablespoons low-sodium soy sauce; 1/3 cup plus 1 tablespoon mirin; 3 tablespoons sake; 2 tablespoons plus 2 teaspoons packed light brown sugar; 3 slices fresh ginger, plus 1 1/2 teaspoons finely grated; 2 bunches scallions, dark green parts separated; 2 tablespoons toasted sesame oil; 1 cup frozen shelled edamame, thawed; 6 ounces soba noodles; 1 1/2 pounds skinless, boneless chicken breasts, cut into 1 1/2-inch cubes; Kosher salt and freshly ground white pepper; Vegetable oil, for brushing