Preheat your oven to 375°F (190°C) and lightly grease a 9x13-inch baking dish.
Melt the butter in a large skillet over medium heat. Add the onion, carrots, celery, and potatoes, then cook for 6 to 8 minutes, stirring occasionally, until the vegetables begin to soften.
Sprinkle the flour over the vegetables and stir for about 1 minute. Slowly pour in the chicken broth while stirring, then add the milk. Keep stirring until the sauce becomes thick and creamy.
Stir in the cooked chicken, frozen peas, thyme, salt, and black pepper. Cook for another 3 to 4 minutes, until everything is hot and well combined.
Transfer the creamy chicken mixture to the prepared baking dish. Arrange the refrigerated biscuits over the top, leaving a little space between each one.
Brush the biscuit tops with melted butter and bake uncovered for 25 to 30 minutes, or until the biscuits are golden brown and the filling is bubbling around the edges. If the biscuits brown too quickly, loosely cover the top with foil.
Let the casserole rest for 5 to 10 minutes before serving so the filling can thicken slightly.
Serve warm with your favorite side dish.