Preheat oven to 400°F (200°C) and grease a 9×13 baking dish.
In a large skillet, melt butter over medium heat. Add carrots, celery, and onion, cooking until softened (about 5 minutes).
Stir in flour and cook for 1-2 minutes to form a roux. Gradually whisk in chicken broth and milk, stirring until smooth and thickened.
Add shredded chicken, garlic powder, thyme, salt, and pepper. Stir to combine, then pour the mixture into the prepared baking dish.
Arrange the biscuits evenly on top of the chicken mixture.
Bake for 20-25 minutes, or until the biscuits are golden brown and cooked through.
Let cool for 5 minutes, garnish with fresh parsley if desired, and serve warm.