Combine the soy sauce, dry vermouth, sake, water, sugar, sesame oil, garlic and ginger in a small saucepan. Bring to a boil, then reduce the heat and simmer for about 15 minutes, until the sauce thickens enough to coat the back of a spoon. Set aside to cool slightly and reserve about ⅓ cup for serving.
Soak 8 to 10 wooden skewers in water for about 30 minutes. Place the chicken pieces in a large bowl and pour the marinade over the top, tossing well to coat. Cover and refrigerate for a couple of hours if time allows.
When ready to cook, thread the chicken onto the skewers. If grilling, lightly oil the grates and cook the skewers for about 8 minutes, turning occasionally, until lightly charred and the internal temperature reaches 165°F. For the oven, preheat to 450°F and roast the skewers for 12 to 15 minutes, until cooked through and caramelized at the edges. Garnish with sliced scallions and serve with the reserved sauce.
