Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat. Add the diced onion, minced garlic, carrots, and celery. Sauté for about 5 minutes until the vegetables are softened.
Add the uncooked rice to the skillet and stir to combine with the vegetables.
Pour in the chicken broth and bring the mixture to a simmer.
In a large casserole dish, spread the rice and vegetable mixture evenly.
Place the chicken pieces on top of the rice mixture.
Sprinkle dried thyme, dried parsley, salt, and black pepper evenly over the chicken and rice.
Cover the casserole dish tightly with aluminum foil and bake in the preheated oven for 40 minutes.
While the casserole is baking, prepare the dairy-free sauce: In a small bowl, whisk together the coconut milk, nutritional yeast, cornstarch, and water until smooth.
After 40 minutes, remove the casserole from the oven and carefully remove the foil.
Pour the coconut milk sauce evenly over the casserole.
Return the casserole to the oven, uncovered, and bake for an additional 5 minutes until the sauce thickens and the chicken is cooked through.
Remove from the oven and let the casserole rest for 5 minutes before serving.