Prepare the pastry dough in advance and allow it to chill for a minimum of two hours. You may refrigerate the dough overnight if more convenient. You will need both discs for the base and top crusts.
In a large saucepan, place the cubed chicken, sliced carrots, and celery. Pour in just enough water to submerge the ingredients. Bring to a rolling boil over medium-high heat and continue to boil for 10 minutes. Remove from heat, drain thoroughly, and set the mixture aside.
In a separate skillet set over medium heat, melt the butter and sauté the onions and garlic until the onions are soft and translucent and the butter takes on a light golden hue. Add the flour, salt, pepper, and thyme, whisking continuously. Slowly incorporate the chicken broth and the half-and-half. Continue to whisk until the mixture is smooth and lump-free, then reduce heat slightly and simmer until the mixture thickens into a hearty gravy, about 10 minutes. Adjust seasoning if desired. Remove from heat and set aside.
Preheat your oven to 425°F (218°C).
Roll one of the chilled dough discs on a lightly floured surface into a 12-inch circle, rotating as needed to maintain an even thickness. Gently fit the rolled dough into a 9-inch pie dish with a depth of 1.5 to 2 inches. Press the dough evenly into the bottom and along the sides. Trim any overhang with kitchen shears or a sharp knife and discard.
Layer the pre-cooked chicken and vegetable mixture evenly into the pie shell. Scatter the frozen peas over the top. Spoon the thickened gravy over the filling, spreading it evenly to ensure full coverage.
Roll out the second dough disc to the same size and gently place it over the filling. Trim excess dough, then crimp the edges together using a fork to seal. Using a paring knife, make a few small slits in the top crust to allow steam to escape during baking. Brush the surface and edges of the crust with the prepared egg wash.
Bake the pie in the preheated oven for 32 to 38 minutes, or until the top crust has turned a deep golden brown. After 20 minutes of baking, shield the edges with foil or a crust protector to avoid overbrowning.
Once baked, remove the pie from the oven and allow it to rest for at least 10 minutes before slicing and serving. The filling will set further upon cooling and is excellent the next day. Store leftovers in an airtight container in the refrigerator for up to 5 days.