In Tokyo, these savory skewers are made with Japanese leeks, called negi, which are thinner and more tender than Western leeks. Here they are substituted with scallions, since they have a similar sweetness and are more readily available.
Sauce can be made 2 days ahead and cooled, uncovered, then chilled, covered.
Line bottom of a broiler pan with foil and cover with rack. Lightly brush rack with oil.
Bring mirin, soy sauce, sake, and sugar to a boil in a 1-quart saucepan over moderately high heat, stirring until sugar is dissolved.
Boil, uncovered, until reduced to about 1/3 cup, about 5 minutes. Remove from heat and reserve 1 1/2 tablespoons sauce in a small bowl for brushing after skewers are cooked.
Cut white and pale green parts of scallions crosswise into 1 1/2-inch pieces (reserve scallion greens for another use).
Cook scallions in a 2-quart saucepan of boiling (unsalted) water 1 minute, then drain in a colander. Immediately transfer to a bowl of ice and cold water to stop cooking. Drain, then pat scallions dry with paper towels.
Thread 1 skewer alternately with chicken and scallions, using 3 pieces of each and piercing scallions crosswise through center.
Place skewer on rack of broiler pan, then brush both sides generously with sauce. Repeat with remaining skewers, arranging all skewers in same direction. (You may have some chicken and scallion pieces left over.)
Cover exposed skewers (but not chicken or scallions) with a strip of foil to prevent scorching.
Broil skewers 4 to 6 inches from heat until chicken is slightly charred, about 4 minutes.
Remove foil and brush both sides with more sauce, then turn skewers over and replace foil. Broil until other side of chicken is slightly charred and cooked through, about 4 minutes more.
Serve warm or at room temperature.
Just before serving, coat skewers with reserved sauce using cleaned brush.
Skewers can be assembled, but not brushed with sauce, 1 day ahead and chilled, covered. Brush with sauce before broiling as directed.