Prepare Pie Crust (2) per your favorite recipe or simply thaw if frozen.
After the pie dough has chilled, on a floured work surface, roll out 1 half of the chilled pie dough. Turn the pie crust dough about a quarter turn after every few rolls until you have a circle 12 inches in diameter.
Generously sprinkle the bottom of your 9-inch deep dish glass place with flour (this will prevent the crust from sticking to the bottom of your pie place when serving). Carefully place the dough into the prepared pie plate. Tuck it in with your fingers, making sure it is smooth. Refrigerate while you are preparing the filling.
In a large saucepan, combine Yukon Gold Potato (1 cup), Carrot (2), and Celery (1). Add water to cover and bring to a boil; reduce heat and simmer for 12 minutes or until the vegies are tender.
Remove from heat, drain, and set aside.
In a large skillet, cook the Unsalted Butter (1/3 cup), Onion (1/3 cup), and Garlic (1 teaspoon) over medium heat, stirring occasionally. Cook until the onions are translucent and the butter is lightly browning.
Sprinkle with All-Purpose Flour (1/3 cup) and stir until you have a barely light brown roux (closer to blonde).
Whisk in the Chicken Broth (1 3/4 cup), Half and Half (2/3 cup), Chicken Bouillon Powder (1 tablespoon), Ground Black Pepper (1/2 teaspoon), and Dried Thyme (1/2 teaspoon). Continue cooking and stirring until no flour lumps remain.
Simmer over medium-low heat until rich and thick, about 10 minutes or slightly longer. Taste and adjust seasonings if you prefer.
Remove from heat and stir in Chicken Breast (3 cup), cooked veggies, and Frozen Green Peas (1 cup); set aside.
Preheat oven to 425 degrees F (220 degrees C).
Roll out second pie crust dough just as you did the first.
Gently pour the chicken mixture into the crust. Cover the pie with the second crust and trim the extra overhang off the sides with a small sharp knife. Seal the edges by crimping with your fingers or a fork. Slice a few small slits in the top crust for steam to escape.
In a small bowl, beat the Egg (1) with a fork. Using a pastry brush, brush crust top and edges with beaten egg.
Bake for 15 minutes, and then place a pie crust shield on top to protect the edges from browning too much.
Continue baking for about 25-30 minutes, or until crust is golden brown.
Remove from oven, remove pie shield and let sit for 10 minutes before serving – this will allow your filling to thicken slightly.
Garnish with some Fresh Thyme (to taste) and serve warm.