Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat.
Add diced onion and cook for 3-4 minutes until translucent.
Add minced garlic, sliced carrots, celery, and mushrooms; cook for 5-6 minutes until vegetables are tender.
Sprinkle gluten-free flour over the vegetables and stir well to coat; cook for 1-2 minutes to remove raw flour taste.
Gradually whisk in the gluten-free chicken broth and whole milk, stirring constantly until the sauce thickens, about 3-4 minutes.
Add shredded cooked chicken, frozen peas, dried thyme, dried parsley, salt, and black pepper; stir to combine and heat through for 2 minutes.
Transfer the mixture to a greased 9×13 inch baking dish.
In a small bowl, mix gluten-free breadcrumbs with shredded cheddar cheese.
Sprinkle the breadcrumb and cheese mixture evenly over the casserole.
Bake uncovered in the preheated oven for 25-30 minutes or until the top is golden brown and the casserole is bubbly.
Remove from oven and let rest for 5 minutes before serving.