Combine everything except chicken and skewers in a large nonreactive bowl and whisk to combine. Add chicken strips and toss to coat. Cover and marinate at least 4 hours.
When ready to cook, heat oven to 425 degrees and arrange a rack in the middle. Remove chicken from marinade (reserving marinade) and thread 1 piece of chicken on each skewer.
Heat the reserved marinade in a saucepan until boiling and boil for 3 minutes. Cool a bit before using.
Arrange skewers on a foil-lined baking sheet in a single layer. Bake for 10 minutes, flip skewers and baste with remaining marinade (you don't need to use it all). Bake about 5 more minutes, until golden brown on both sides. Transfer to a platter, garnish skewers with scallions, and serve.