Preheat your oven to 400°F (200°C) and lightly grease a 9×13-inch baking dish. Melt the butter in a large skillet over medium heat. Add the diced onions, celery, and carrots, then cook for about 6 to 8 minutes until softened. Stir in the minced garlic and cook for another minute until fragrant.

Sprinkle the flour over the cooked vegetables and stir continuously for about 2 minutes to eliminate the raw flour taste. Slowly whisk in the chicken broth until smooth, followed by the milk and heavy cream. Continue stirring until the sauce thickens into a rich, creamy consistency. Add the thyme, parsley, garlic powder, onion powder, salt, and black pepper.

Fold the shredded cooked chicken and frozen peas into the creamy sauce. Stir gently until everything is evenly coated and heated through. Taste the filling and adjust the seasoning if necessary.

Pour the creamy chicken mixture into the prepared baking dish and spread it into an even layer. Place the pie crust or puff pastry over the top, trimming any excess if needed. Cut a few small slits in the crust to allow steam to escape during baking. Brush the top evenly with beaten egg for a beautifully golden finish.

Bake the casserole for 30 to 35 minutes, or until the topping is deep golden brown and the filling is bubbling around the edges. If the crust begins browning too quickly, loosely cover it with aluminum foil during the final 10 minutes of baking.

Remove the casserole from the oven and let it rest for about 10 minutes before serving. This allows the creamy filling to set slightly, making it easier to scoop neat portions. Garnish with freshly chopped parsley and freshly cracked black pepper if desired. Serve warm and enjoy every bite of the flaky golden crust and creamy chicken filling.