Preheat oven to 375°F (190°C).
In a large skillet, melt butter over medium heat. Add chopped onion and minced garlic; sauté for 3-4 minutes until softened.
Stir in flour and cook for 1 minute, stirring constantly to form a roux.
Gradually whisk in chicken broth and milk, continuing to whisk until sauce thickens, about 5 minutes.
Add dried thyme, salt, and black pepper to the sauce; stir to combine.
In a large mixing bowl, combine shredded cooked chicken, broccoli florets, diced carrots, sliced mushrooms, frozen peas, cooked rice, and the prepared sauce. Mix well.
Transfer the mixture into a greased 9×13 inch baking dish and spread evenly.
Sprinkle shredded cheddar cheese evenly over the top.
In a small bowl, mix panko breadcrumbs with olive oil until coated. Sprinkle the breadcrumb mixture over the cheese layer.
Cover the casserole with aluminum foil and refrigerate if preparing ahead of time. To bake immediately, place the covered casserole in the preheated oven.
Bake for 30 minutes covered. Remove the foil and bake for an additional 15 minutes until the top is golden brown and bubbly.
Remove from oven and let rest for 5 minutes before serving.