Preheat your oven to 350 degrees and spray a nine-inch square baking pan with cooking spray.
First, you will prepare the teriyaki sauce. In a medium-sized pot, combine the soy sauce, water, and brown sugar, honey ginger, oil, and garlic. Use a whisk to combine them and put them on the stove over medium-high heat.
As the mixture is heating up, add the water and the flour to a small jelly jar with a lid. Put the lid on the jar and shake the jar to combine the two ingredients. If you don’t have a jelly jar, you can simply whisk the flour into the water in a small bowl.
When the pot comes to a boil, let it boil for about 60 seconds. Take a spoonful of the hot mixture and stir it into the flour mixture to raise the temperature of the flour mixture a bit.
Then, pour the flour and water mixture into the pot of seasonings, whisking as you do so. Boil the mixture until it begins to thicken, whisking and stirring to keep it from forming lumps. Remove the pot from the heat and set it aside.
Put the chicken breasts in the prepared pan and top them with about a cup of the sauce mixture.
Bake the chicken for about half an hour, and when you remove it from the oven, use two forks to shred it into small pieces.
While the chicken is baking, prepare the frozen stir fry vegetables according to package instructions. Also, while the chicken is baking, you can prepare the rice according to the package instructions.
Add the rice and vegetables to the pan that contains the shredded chicken and stir to combine them. Top the mixture with several tablespoons of the teriyaki sauce, stirring the sauce into the mixture.
Return the pan to the oven and bake it for about 15 more minutes. Drizzle the top of the casserole with more sauce and serve the casserole immediately in bowls.