Preheat your oven to 400°F. Whisk the egg with 1 tablespoon of water in a small bowl and set aside for the wash.

In a medium saucepan, melt the butter over medium-low heat. Add the diced onion and cook, stirring occasionally, until translucent and tender, about 5 minutes. Stir in the flour, poultry seasoning, and thyme, and season with salt and black pepper. Continue cooking for 2 minutes, stirring constantly, to make a roux.

Pour in the chicken broth and milk gradually, whisking constantly after each addition. The sauce will look very thick at first, then become smooth. Bring to a simmer over medium heat and cook for 1 minute, then remove the saucepan from the heat. Stir in the turkey, frozen vegetables, and cubed potato until well combined.

Line a 9-inch pie plate with one of the pie crusts. Fill with the turkey mixture. Brush the rim of the bottom crust with egg wash, then place the second crust on top. Press the edges together to seal, trim away excess dough, and cut a few slits in the top to vent. Brush the entire top crust with the remaining egg wash.

Place the pie on a rimmed baking sheet and bake until the crust is golden brown and the filling is bubbling, about 35 to 40 minutes. If the edges brown too quickly, tent loosely with foil.

Transfer the pie to a wire rack and cool for 10 to 15 minutes before slicing. This helps the filling set and makes clean slices.