Preheat oven to 350° F. Spray a 9 x 13-inch baking dish with cooking spray and set aside.
Melt ⅓ cup butter in a Dutch oven over medium heat. Add the onion and saute until soft, about 4-5 minutes. Add mushrooms, carrot and celery. Cook until vegetables soften, about 5-6 minutes. Add garlic; cook for 1 more minute, stirring constantly. Add the flour, salt and pepper; cook, stirring constantly, for 1 more minute.
Gradually whisk in the broth; bring to a boil. Reduce heat slightly, and let the sauce bubble and simmer until it thickens, about 2-3 minutes. Reduce heat to low, stir in the milk; add rice, chicken, parsley and thyme. Cook and stir for 1-2 more minutes. Taste and season with additional salt and pepper, if necessary. Remove from heat.
Transfer chicken and rice mixture to the prepared baking dish.
In a small bowl, stir together the Ritz cracker crumbs and 1 tablespoon of melted butter. Sprinkle the buttered crumbs on the top of the casserole.
Bake, uncovered, for 20-25 minutes, or until hot and bubbly.