Soak six 10-inch bamboo skewers in water for 30 minutes to prevent burning.
In a small saucepan, combine tamari, mirin, sake, brown sugar, garlic, and 1/4 cup water. Bring to a boil over medium-high heat.
Stir frequently as the sauce reduces by half, about 12-15 minutes. Remove from heat, stir in the ginger, and let it cool.
Reserve 1/4 cup of the sauce for glazing and set aside the remaining for serving. (You can refrigerate the sauce for up to one day if preparing ahead.)
Preheat a grill or grill pan to medium-high heat. Lightly oil the hot grates with vegetable oil to prevent sticking.
Thread the chicken and scallions onto the soaked skewers, alternating between the two. Aim for 7 to 8 pieces per skewer.
Sprinkle the skewers lightly with kosher salt and place them on the grill.
Cook the skewers, turning frequently, for approximately 12 minutes, until the chicken is browned and fully cooked, and the scallions are tender. The chicken should reach an internal temperature of 165°F.
Brush the reserved 1/4 cup glaze over the skewers and grill for 3-4 minutes more, turning continuously, until the glaze is caramelized.
Serve hot alongside the remaining sauce for dipping.