In a saucepan, bring the water and kombu to a simmer. Add the bonito and return to a simmer. Remove the pan from the heat and let stand for 3 minutes. Strain the broth into a large saucepan. Add the sake, soy sauce, mirin and sugar to the broth and boil over moderately high heat, stirring occasionally, until slightly thickened, 45 minutes.
Meanwhile, soak 16 bamboo skewers in water for 20 minutes; drain. Alternately thread the chicken and scallions onto the skewers.
Finely grate the ginger into a small, fine strainer set over a bowl. Press the juice from the ginger; you should have 2 tablespoons.
Light a grill. Brush the chicken and scallions with oil and grill over moderate heat, turning, until just cooked through, 10 minutes. Just before removing them from the grill, brush the skewers with the ginger juice and soy glaze. Serve, passing more glaze for dipping.