Preheat the Oven: Preheat your oven to 400°F (200°C) and line a large sheet pan with parchment paper or lightly grease with cooking spray.
Prepare the Filling: In a large skillet or saucepan, melt the butter over medium heat. Add the diced onion, carrots, and celery, and sauté until softened, about 5-7 minutes. Add the minced garlic and cook for an additional minute until fragrant.
Sprinkle the flour over the vegetables and stir to coat evenly. Cook for 1-2 minutes, stirring constantly, to cook out the raw flour taste.
Gradually pour in the chicken broth and milk, stirring constantly to prevent lumps from forming. Bring the mixture to a simmer and cook until thickened, about 5 minutes.
Stir in the cooked chicken, frozen peas, dried thyme, dried parsley, salt, and pepper. Cook for another 2-3 minutes, then remove from heat. Adjust seasoning to taste if necessary.
Assemble the Pot Pie: Transfer the chicken and vegetable mixture to the prepared sheet pan, spreading it out evenly.
Add the Crust: Unroll the thawed puff pastry sheet and place it over the filling on the sheet pan, tucking in any excess pastry around the edges. Use a sharp knife to make a few slits in the pastry to allow steam to escape during baking.
Brush with Egg Wash: Lightly brush the beaten egg over the surface of the puff pastry for a golden, shiny finish.
Bake: Place the sheet pan in the preheated oven and bake for 25-30 minutes, or until the crust is golden brown and the filling is bubbly and heated through.
Serve: Once baked to perfection, remove the sheet pan from the oven and let the Chicken Pot Pie cool slightly before slicing and serving.
