Preheat the oven to 375°F. Grease a deep 9x13-inch baking dish with butter or nonstick spray.
Melt 2 tablespoons butter in a large skillet over medium heat. Add the onion and celery and cook for 4 to 5 minutes, until softened.
Add the garlic and cook for 30 seconds, stirring constantly. Remove the skillet from the heat.
In a large mixing bowl, whisk together the cream of chicken soup, cream of celery soup, chicken broth, milk, sour cream, salt, pepper, thyme, and poultry seasoning until smooth.
Stir in the uncooked rice, shredded chicken, frozen peas and carrots, sautéed onion and celery, and 1 cup of the shredded cheddar cheese.
Pour the mixture into the prepared baking dish and spread it into an even layer.
Cover the baking dish tightly with foil. Bake for 50 minutes.
Carefully remove the foil and stir the casserole, making sure to scrape around the edges and redistribute the rice. The rice should be partially tender, and the casserole should still look creamy.
Sprinkle the remaining 1/2 cup cheddar cheese over the top.
In a small bowl, stir together the crushed crackers and melted butter. Sprinkle the buttery cracker crumbs evenly over the casserole.
Return the casserole to the oven, uncovered, and bake for 15 to 20 minutes, until the rice is tender, the filling is bubbling around the edges, and the topping is golden brown.
Let the casserole rest for 10 minutes before serving. Sprinkle with fresh parsley, if desired.