Make the tare (sauce): In a small saucepan, combine the soy sauce, mirin, sake, brown sugar, ginger, garlic, and dark green scallions. Bring to a gentle boil, then reduce the heat and simmer, uncovered, for 5 minutes. Using a slotted spoon, remove and discard the ginger, garlic, and scallions. Stir in the cornstarch slurry and cook for 1 to 2 minutes more, until the sauce thickens enough to coat the back of a spoon. Reserve 1 cup of the sauce for serving and set the rest aside for basting.

Prepare the skewers: Thread the chicken pieces and the light green scallion sections (widthwise) onto the skewers, alternating chicken and scallion pieces.

Cook the yakitori: To broil: Line a baking sheet with heavy-duty foil. Arrange the skewers in a single layer. Broil, 4 to 6 in (10 to 15 cm) from the heat source, for 5 minutes on each side, then brush the top side with sauce. Continue broiling for 1 to 2 minutes, until the chicken is caramelized in spots. Flip, brush with sauce, and broil for another 1 to 2 minutes, until the chicken is glazed and cooked through. To grill: Preheat a grill to medium-high heat and lightly oil the grates. Grill for 4 to 5 minutes per side, then baste with sauce and continue cooking for another few minutes, basting once or twice more, until the chicken is cooked through and caramelized in spots.

Serve: Transfer the skewers to a platter, drizzle with a little of the reserved serving sauce, and sprinkle with sesame seeds, if using. Serve hot with any remaining serving sauce for dipping.