1. Combine first 4 ingredients in a small saucepan; bring to a boil. Remove from heat; let stand 15 minutes. Remove ginger with a slotted spoon, and place in a medium bowl. Add 2 tablespoons soy sauce mixture, pepper, chicken, garlic, and cilantro to ginger in bowl; toss well. Cover and chill 1 hour, stirring occasionally.
2. Preheat grill to medium-high heat.
3. Bring remaining soy sauce mixture to a boil; cook 10 minutes or until reduced to 1/4 cup. Remove from heat; slowly add vinegar and oil, stirring with a whisk. Divide soy sauce mixture into 2 equal portions (about 1/4 cup each); set aside.
4. Remove chicken from marinade; discard marinade. Thread 3 chicken pieces and 3 green onion pieces alternately onto each of 12 (8-inch) skewers. Place the skewers on a grill rack coated with cooking spray. Cover and grill for 8 minutes or until done, turning after 4 minutes and basting occasionally with 1/4 cup reserved soy sauce mixture.
5. Place skewers on a serving platter; serve with remaining 1/4 cup soy sauce mixture.