Preheat the oven to 375 F.
Melt the butter in a large pot over medium-high heat, then add the onion, carrots, and celery. Stir them around until the onions start to turn translucent, about 3 minutes.
Stir in the chicken or turkey and then sprinkle the flour over the top and stir it until it’s all combined with the turkey and vegetables. Cook for 1 minute, then pour in the chicken broth (and wine if using) and stir it around and let it cook and thicken.
Once it starts to thicken add the turmeric, salt, pepper, and thyme.
Add the half-and-half or cream, then stir the mixture and let it bubble up and thicken, about 3 minutes. If it seems overly thick, splash in a little more broth. Turn off the heat.
Pour the filling into a 2-quart baking dish. Roll out the pie crust on a floured surface and lay it over the top of the dish. Press the dough so that the edges stick to the outside of the pan. Use a knife to cut little vents here and there in the surface of the dough.
Mix together the egg with 2 tablespoons water and brush it all over the surface of the crust. (You will have some egg wash left over.)
Place the pie on a rimmed baking sheet and bake for 25 to 30 minutes, until the crust is deep golden brown and the filling is bubbly. To prevent the crust from getting too brown, you might want to cover it lightly with foil for the first 15 minutes of baking time.
Serve up servings by the (big ol’) spoonful!