In the bottom of a 10 inch cast iron skillet, melt the butter over medium heat. Sauté the chopped onion and garlic until translucent; about 2 minutes. Stir in the carrots and celery and cook until the celery is bright green, another 5 minutes.
Add in the chopped potatoes and turkey and cook until potatoes are just starting to soften. About 10 minutes. Stir in the flour and let cook for 2 minutes.
Deglaze pan with white wine, using your utensil to scrape off any stuck on bits from the bottom of the pan. Stir in the chicken stock, rosemary, thyme, and parsley and bring to a simmer for 5 minutes.
Preheat oven to 400°F. Remove the skillet from the stovetop, stir in the frozen peas and season with salt and pepper.
Place the prepared pie crust on top of the skillet, tucking the edges under as needed. Cut a slit or decoration in the top for the steam to release. Mix together the egg and milk and brush it over the top of the pie dough.
Put the skillet into the preheated oven bake for 30 minutes, or until the crust is golden brown and the filling begins to bubble out of the slit.
Remove from the oven and let rest for 10 minutes before serving.