Preheat the oven to 350°F (175°C).
In a large skillet, melt the butter over medium heat. Add the diced onion, carrots, celery, and mushrooms. Cook for 5-7 minutes until vegetables are softened.
Add the minced garlic and cook for an additional 1 minute until fragrant.
Sprinkle the flour over the vegetables and stir well to combine. Cook for 2 minutes to remove the raw flour taste.
Gradually whisk in the chicken broth and milk, stirring constantly to avoid lumps. Bring the mixture to a simmer and cook for 5 minutes until thickened.
Stir in the dried thyme, dried parsley, salt, and black pepper.
In a large mixing bowl, combine the shredded cooked chicken, steamed broccoli florets, cooked white rice, and the creamy vegetable sauce. Mix well.
Transfer the mixture to a greased 9×13 inch casserole dish. Sprinkle the shredded cheddar cheese evenly over the top.
Sprinkle the plain breadcrumbs over the cheese layer for a crunchy topping.
Cover the casserole with aluminum foil and refrigerate if making ahead. When ready to bake, remove from refrigerator and let sit at room temperature for 15 minutes.
Bake covered for 30 minutes. Remove the foil and bake for an additional 15 minutes until the top is golden brown and bubbly.
Let the casserole rest for 5 minutes before serving.