Preheat the oven to 350 degrees F. Lightly coat a 9x13-inch (3-quart) casserole dish with cooking spray. If necessary, cook the brown rice.
In a small saucepan over medium heat, combine the soy sauce, 1/2 cup water, rice vinegar, honey, brown sugar, garlic, and ginger. Cover and bring to a boil. Once boiling, remove the lid and cook for 1 minute, stirring constantly. While you wait for the sauce to boil, whisk the cornstarch and 2 tablespoons of water together in a small bowl to create a slurry. Once the sauce is boiling, add the slurry, then stir to combine. Cook for 1 additional minute, until the sauce begins to thicken. Remove from heat.
Arrange the chicken breasts in a single layer in the prepared casserole dish. Pour 1 cup of the sauce over the top, then bake the chicken for 30 to 35 minutes, until cooked through. Remove the dish from the oven and shred the chicken in the dish with two forks.
While the chicken cooks, steam the vegetables according to package instructions. Add the vegetables, brown rice, and edamame to the dish with the shredded chicken. Add all but 1/4 cup of the remaining sauce. Stir together, then return to the oven and bake until heated through, about 15 minutes. Remove from the oven, drizzle with reserved sauce, and sprinkle with green onions. Enjoy warm.
1 cup of uncooked long grain brown rice yields 3 1/2 to 4 cups cooked. Thus, if you cook 1 full cup of rice, you will have a little extra leftover, which you can save for a different recipe or go ahead and mix into the casserole anyway—just know that the proportions of rice to chicken/veggies/sauce and nutrition info will be different. If using white rice, 1 cup uncooked long grain white yields about 3 cups cooked.
Store leftovers in an airtight container in the refrigerator for up to 5 days or freeze for up to 2 months. Let thaw overnight in the refrigerator.
If reheating in the microwave, I recommend topping your serving with a small splash of water or chicken stock to keep it from drying out.