Heat oil and sauté onions, garlic, and vegetables until softened, ensuring they release flavor and cook evenly before baking.
Prepare a roux with butter and flour, then gradually whisk in broth and milk to create a smooth, thick sauce.
Combine cooked chicken and vegetables with the sauce, allowing everything to simmer briefly for flavor blending.
Transfer the mixture into a baking dish and cover with pastry or biscuit topping, ensuring even coverage.
Bake until the top is golden and the filling is bubbling, indicating the casserole is fully cooked.
Allow the dish to rest before serving so the sauce thickens and holds its structure when plated.