Heat the oven to 400°F. Beat the egg together with 1 tablespoon of water in a small bowl to make an egg wash and set aside.

Melt the butter in a medium saucepan over medium-low heat. Add the chopped onion and cook until soft and translucent, about 5 minutes. Stir in the flour, poultry seasoning, dried thyme, plus a pinch of salt and black pepper, and cook for 2 minutes, stirring constantly.

Slowly whisk in the chicken broth and milk, a splash at a time, whisking until smooth after each addition. The sauce will look very thick at first but will loosen as you add more liquid. Bring to a simmer over medium heat and cook for 1 minute, then remove from the heat. Fold in the chopped turkey, frozen vegetables, and diced potato.

Place one of the pie crusts into a 9-inch pie plate, easing it into the bottom and sides. Spoon the turkey filling into the crust. Brush the rim of the bottom crust with the egg wash, then lay the second crust on top. Press the edges together to seal, then crimp with your fingers or a fork. Cut 4 or 5 slits in the top crust for steam, and brush the entire surface with the remaining egg wash.

Bake for 35 to 40 minutes, until the crust is deep golden brown and the filling is hot and bubbling. If the edges brown too quickly, cover the pie loosely with foil.

Let the pie cool on a wire rack for 10 to 15 minutes before slicing. This lets the filling settle so each slice holds together.