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 45 minutes and up to 4 hours.
When ready to cook, heat oven to 425 degrees F 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 to 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.
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