Begin by preheating your oven to 400°F. In a medium saucepan over medium heat, melt the butter. Once it has melted, stir in the chopped onion and cook for about 2 minutes, stirring occasionally to prevent sticking. Add the minced garlic and cook for an additional 30 seconds, stirring constantly.

Sprinkle the flour into the pan, whisking it with the butter, garlic, and onion to form a smooth roux. Let this cook for about 30 seconds, still whisking continuously.

Gradually pour in the chicken broth and milk, whisking constantly to combine with the flour mixture. Bring the sauce to a boil and allow it to simmer for 2 minutes, stirring to avoid lumps.

Remove the pan from heat and season the sauce with garlic powder, onion powder, salt, black pepper, and thyme. Stir well to incorporate the spices.

Fold in the frozen peas, carrots, and shredded chicken into the sauce. Make sure the vegetables and chicken are evenly distributed throughout the mixture.

Transfer the chicken and vegetable filling to an 8x8-inch casserole dish. If needed, gently roll out the puff pastry, sealing any seams, until it fits over the top of the dish with about half an inch of excess dough around the edges. Place the pastry over the filling, and make four small slits in the top to allow steam to escape during baking.

In a small bowl, whisk together the beaten egg with a tablespoon of water. Lightly brush the egg wash over the surface of the puff pastry to encourage a golden finish.

Bake in the preheated oven for 25 to 30 minutes, or until the puff pastry is beautifully golden and crisp. If desired, garnish with minced parsley before serving. Enjoy your homemade chicken pot pie with its flaky puff pastry crust!