Preheat the oven to 375°F (190°C) and lightly grease a 9x13-inch baking dish.
Melt the butter in a large skillet over medium heat. Add the onion and cook for 4 to 5 minutes until softened.
Stir in the garlic and cook for 30 seconds until fragrant.
Sprinkle the flour over the onion mixture and cook for 1 minute, stirring constantly.
Gradually whisk in the chicken broth and milk until smooth. Add the garlic powder, onion powder, dried thyme, salt, and black pepper. Simmer for 3 to 5 minutes until the sauce begins to thicken.
Remove the skillet from the heat and stir in 1 cup of the cheddar cheese and the Parmesan cheese until melted and smooth.
Fold in the cooked chicken, cooked rice, and frozen peas until evenly coated in the sauce.
Transfer the mixture to the prepared baking dish and sprinkle with the remaining cheddar cheese.
Bake for 25 to 30 minutes, or until the casserole is hot and bubbly and the cheese is melted. Broil for 2 to 3 minutes for a golden topping if desired.
Let the casserole rest for 10 minutes before garnishing with fresh parsley and serving.