Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat.
Add diced onion and minced garlic; sauté for 3-4 minutes until softened.
Add sliced carrots, celery, and mushrooms to the skillet; cook for 5-6 minutes until vegetables begin to soften.
Sprinkle gluten-free flour over the vegetables and stir well to coat; cook for 1-2 minutes to remove raw flour taste.
Gradually pour in the chicken broth while stirring continuously to avoid lumps.
Add milk, dried thyme, salt, and black pepper; stir and cook until the sauce thickens, about 5 minutes.
Remove the skillet from heat and stir in the cooked chicken and frozen peas.
Transfer the mixture to a 9×13 inch (23×33 cm) casserole dish and spread evenly.
In a small bowl, combine gluten-free breadcrumbs and grated Parmesan cheese.
Sprinkle the breadcrumb mixture evenly over the casserole.
Bake in the preheated oven for 30 minutes or until the top is golden brown and the casserole is bubbly.
Remove from oven and let it rest for 5 minutes before serving.