In a wide dish, toss the chicken pieces in the teriyaki marinade. Cover and refrigerate at least 30 minutes to 12 hours.

Preheat oven to 400 degrees If using, place a baking stone on the top rack in the center of the oven.

In a large skillet, bring chicken with marinade to a simmer over medium heat. Cook, stirring occasionally, until pieces are no longer pink in center. Use a slotted spoon to remove chicken to a plate.

In a small bowl, whisk together cornstarch with 3 tablespoons of the marinade from the skillet. Stir the mixture into the teriyaki and continue to cook, stirring constantly, until thickened, about 5 minutes. Remove from heat.

If using a baking stone, place par-baked crust on a pizza peel sprinkled with some cornmeal. Otherwise, place the crust on a baking sheet. Spread the thickened teriyaki over the pizza, leaving a 1/2 inch border around the edges. Evenly cover with cabbage, then carrots. Top with the cooked chicken pieces, then Mozzarella and Cheddar cheeses.

Transfer to pizza stone or bake on baking sheet until crust is golden and cheese is melted and bubbly, about 10 minutes. Let sit for 10 minutes before slicing and serving.