Preheat the oven to 375°F (190°C).
In a large mixing bowl, combine the shredded turkey, mashed potatoes, diced carrots, chopped green beans, peas, and turkey gravy.
Add dried thyme, black pepper, and salt to the mixture and stir until well combined.
Melt the butter in a skillet over medium heat. Pour the turkey mixture into the skillet and cook for 3-4 minutes, stirring occasionally, to warm through and blend flavors. Remove from heat.
Roll out one sheet of the pie crust and fit it into a 9-inch pie dish, pressing gently to cover the bottom and sides.
Pour the turkey and vegetable filling into the pie crust, spreading evenly.
Roll out the second sheet of pie crust and place it over the filling. Trim any excess dough and crimp the edges to seal the pie.
Cut a few small slits in the top crust to allow steam to escape during baking.
Brush the top crust with the beaten egg to give it a golden color when baked.
Place the pie on a baking sheet to catch any drips and bake in the preheated oven for 35-40 minutes, or until the crust is golden brown and the filling is bubbling.
Remove the pie from the oven and let it cool for 10 minutes before slicing and serving.