Preheat oven to 350 degrees F. In an oven-safe 4 quart casserole pan, melt butter over medium heat. Add the onions and carrots and saute until the onions are soft, about 5 minutes. Add celery and mushrooms and continue to saute for 3-5 minutes.

In a small bowl, combine flour, salt, pepper, and thyme. Add the flour mixture to the cooking vegetables. Cook one minute, stirring the entire time. Pour in the broth and half-and-half. Bring the sauce to a boil and continue to stir until sauce thickens. Stir in 1/2 the parmesan cheese.

Remove from heat and fold in the asparagus, wild rice, brown rice, and chicken. Sprinkle the top with remaining cheese.

Cover the pan and bake for 15-20 minutes until the sauce is bubbly. Remove the cover and bake for 5-10 minutes more, until the top of the casserole is light golden. Allow casserole to set for 5-7 minutes before serving.