Preheat the oven to 350°F. Lightly grease a 9x13-inch baking dish with nonstick cooking spray.
Melt the butter in a large skillet over medium heat. Add the diced onion and celery and cook for about 7–10 minutes, stirring occasionally, until the vegetables are soft and tender. Season lightly with salt and black pepper.
Pour the cooked onion, celery, and any remaining melted butter into the bottom of the prepared baking dish. Spread everything into an even layer.
Arrange the shredded chicken evenly over the vegetables. Rotisserie chicken works especially well for this recipe and makes the casserole quick and easy to prepare.
In a medium bowl, stir together the Bisquick and milk just until combined. Carefully pour the Bisquick mixture evenly over the chicken. Do not stir the layers together.
In a separate bowl, whisk together the chicken broth, cream of chicken soup, dried thyme, garlic powder, onion powder, salt, and black pepper until smooth.
Slowly pour the cream of chicken mixture evenly over the Bisquick layer. Do not stir. Keeping the layers separate allows the Bisquick mixture to bake into a soft dumpling layer while the chicken broth and soup create a creamy gravy underneath.
Bake the chicken and dumplings casserole uncovered for 45–50 minutes, or until the top is golden brown, the edges are bubbling, and the dumpling layer is cooked through.
Remove the casserole from the oven and allow it to rest for 5–10 minutes before serving. The creamy chicken filling will thicken slightly as the casserole rests.
Garnish with fresh chopped parsley, if desired, and serve warm.