Set the oven to preheat at 375º Fahrenheit with the rack placed in the middle.
In a large skillet, melt the butter over medium-high heat, then add diced carrots, celery, onion, fresh thyme, and half of the salt. Stir often and cook until the vegetables become soft, about 5 to 7 minutes.
Sprinkle the flour over the veggies and stir well to coat. Cook another 2 minutes to remove the raw flour taste.
Gradually pour in the milk and chicken broth while whisking. Add Dijon mustard, garlic powder, and the remaining salt, then bring the mixture to a boil over high heat.
Lower the heat to medium and continue stirring frequently until the sauce thickens, roughly 3 to 5 minutes.
Fold in the shredded chicken and frozen peas, then taste and adjust salt if needed.
Transfer the filling into a 13-by-9-inch baking dish. Arrange biscuit dough pieces evenly on top of the filling and sprinkle with black pepper and a little extra salt.
Bake in the oven until the filling bubbles and biscuits turn golden and puffed, about 22 to 25 minutes.
Allow the casserole to rest for 5 minutes before slicing and serving.
