Preheat the oven to 400°F (204°C). Combine the beaten egg with a tablespoon of water to create an egg wash and keep it aside.
Melt the butter in a medium saucepan over medium-low heat. Sauté the diced onion in the melted butter until it turns soft and translucent, around 5 minutes. Incorporate the flour, poultry seasoning, dried thyme, and a pinch of salt and black pepper. Stir continuously and cook for an additional 2 minutes.
Gradually whisk in the chicken broth and milk, a little at a time, ensuring the mixture stays smooth. The sauce will begin thick but will loosen as the liquids are fully integrated. Increase the heat to medium and bring the mixture to a gentle boil. Simmer for 1 minute until thickened. Remove from heat.
Stir the cooked turkey, thawed vegetables, and prepared potato into the thickened sauce, combining thoroughly.
Fit one sheet of the pie pastry into a 9-inch pie plate. Pour in the turkey filling, distributing it evenly. Moisten the edge of the bottom crust with the egg wash, then lay the second crust over the top. Seal the edges by pinching them together, and slice a few small vents into the top crust to release steam during baking. Brush the entire top crust with the reserved egg wash.
Bake in the preheated oven for 35 to 40 minutes or until the crust turns a golden hue and the filling is bubbling.
Allow the pie to rest for 10–15 minutes before slicing and serving.