Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat. Add the diced onion and cook for 3-4 minutes until softened.
Add the minced garlic and cook for 1 minute until fragrant.
Stir in the gluten-free flour and cook for 1-2 minutes, stirring constantly to form a roux.
Gradually whisk in the chicken broth and milk, continuing to whisk until the sauce thickens, about 3-5 minutes.
Add dried thyme, salt, and black pepper to the sauce and stir well.
Add the cooked chicken, broccoli florets, sliced mushrooms, diced carrots, and peas to the sauce. Stir to combine evenly.
Remove the skillet from heat and stir in the shredded cheddar cheese until melted and well incorporated.
Transfer the mixture to a 9×13 inch casserole dish and spread evenly.
In a small bowl, combine gluten-free breadcrumbs with melted butter. Sprinkle this mixture evenly over the casserole.
Bake uncovered for 30-35 minutes or until the topping is golden brown and the casserole is bubbly.
Remove from oven and let cool for 5 minutes before serving.