Soak 10 wooden skewers in cold water for 15 minutes.
Thread chicken pieces onto the soaked skewers, alternating with scallions.
Combine sake, soy sauce, mirin, and sugar in a small saucepan and bring to a boil. Reduce heat and simmer for 5 minutes. Reserve 1/2 of the sauce for dipping.
Heat a grill pan over high heat and lightly brush with vegetable oil. Add skewers and cook until chicken is no longer pink in the center, basting frequently with 1/2 of the sauce, 7 to 10 minutes per side.