In a saucepan, combine the sake, soy sauce, sugar and 1/4 cup of the mirin; boil until reduced to 3/4 cup, 3 minutes. Let cool.
Preheat the oven to 375°. In a bowl, combine the chicken, salt, shallot, zest and the remaining 2 tablespoons of mirin. Lightly coat a rimmed baking sheet with 1 teaspoon of the oil. Form the chicken mixture into 24 meatballs. Brush the meatballs with the remaining 2 teaspoons of oil and arrange them on the baking sheet. Bake for about 6 minutes, until the meatballs are barely cooked through.
Light a grill. Thread the meatballs onto 8 bamboo skewers and grill over moderately high heat, turning, until lightly charred, about 2 minutes. Reduce the heat to low and brush the meatballs with the sauce. Grill, turning and brushing, until glazed, 30 seconds longer. Serve with the remaining sauce.