Preheat the oven to 350 degrees. Lightly mist a 9x13 casserole dish with cooking spray and set aside.
Melt the butter in a skillet over medium heat. Add the onion, celery and carrots and sprinkle with a little bit of salt and pepper. Stir together to coat. Saute the vegetables for 8-10 minutes, stirring occasionally, until softened. Transfer the vegetables to the prepared baking dish and spread in a single layer across the bottom of the dish. Top the vegetables with a layer of the shredded cooked chicken.
In a mixing bowl, combine the self-rising flour and milk and stir together until well combined and smooth. Pour the mixture evenly over the top of the chicken layer. Do NOT mix the layers together!
In another mixing bowl, whisk together the cream of chicken soup, chicken broth, parsley, thyme, salt and pepper until fully combined and smooth. Pour this mixture over the top of the other layers in the baking dish. Do NOT mix or stir the layers!
Place the dish in the oven and bake for 40-45 minutes.