Lay chicken breasts flat between pieces of plastic wrap or waxed paper. Using a meat mallet, pound each breast to 1/4-inch thickness. Slice each breast lengthwise into 3 equal pieces. Cut the green onions into 3-inch long pieces. Arrange 3 green onion sticks down the middle of 1 flattened chicken breast strip. Spirally, roll the chicken tightly around the green onions and secure with kitchen twine. Alternatively, leave scallions whole and roll up chicken. Repeat with remaining chicken breasts and green onions. Set aside.
Combine sake, mirin, soy sauce, sugar, garlic, ginger and sesame oil in a zip-top plastic bag or other sealable container. Add the chicken wraps and marinate for 4 hours.
Preheat the oven to 400 degrees F.
Remove chicken wraps from the marinade, allowing excess marinade to drip off. Reserve marinade, if desired. Heat the vegetable oil in a large, oven-proof skillet over medium-high heat. Add the chicken wraps and cook, turning occasionally, until well browned, 5 to 6 minutes. Pour the marinade over, if desired, and place the skillet in the oven and continue cooking until the chicken breasts are cooked through, 5 to 6 additional minutes.
Transfer negimaki to a cutting board and allow meat to rest for 5 minutes before slicing. Serve whole or slice the negimaki crosswise, into pieces. Serve the Soy Glaze as a dipping sauce.
Combine soy sauce, mirin, and sugar in a small saucepan over high heat. Bring to a boil, stirring constantly until the sugar dissolves. Allow mixture to reduce over high heat until it reaches a syrupy consistence, about 5 to 6 minutes. Remove from the heat and stir in the water. Cool to room temperature.