Whisk 2 tablespoons each: soy sauce, mirin, and sake in a large bowl. Add chicken, toss to coat, and refrigerate for 1 hour.
In a small saucepan, combine remaining 1/4 cup soy sauce, 1/4 cup mirin, and 2 tablespoons each of the sake and sugar. Bring the mixture to a boil, lower the heat, and let it simmer until the glaze is slightly thickened and glossy (about 15 minutes).
Prepare your grill or heat a grill pan to medium high. Lightly oil the grill grate.
Thread 4 chunks of chicken and 3 pieces of scallion onto each skewer, alternating the chicken and scallions.
Grill 10 minutes, or until meat is cooked through, turning every 2 to 3 minutes. Baste with glaze on both sides during the last 3 minutes. Brush with the glaze a second time during the final minute of cooking. Chicken and scallions should be deeply glazed when finished cooking.