Place the bottom pie crust in the prepared deep dish pie plate. Set aside for now.
In a large pot, combine the broth, potatoes, carrots, celery, peas, pearl onions, salt, pepper, poultry seasoning, and nutmeg. Bring to a boil; then reduce to a simmer, cover, and cook 15-20 minutes, 'til the vegetables are tender.
While the vegetables simmer, melt the butter in a separate small saucepan over low to medium heat. Whisk in the flour and cook 1-2 minutes, stirring constantly 'til it forms a smooth paste (or sauce) and smells lightly toasted. This is your roux.
Once the vegetables are tender, ladle a cup or so of the hot broth into the roux, whisking to combine.
Then pour the roux mixture back into the vegetables, along with the 1/2 cup milk, whisking to combine.
Simmer, stirring occasionally, 'til the filling has thickened to a gravy-like consistency. It will take just a few minutes.
Remove the pot from the heat. Fold in the turkey and mashed egg.
Pre-heat the oven to 375° F.
Pour the filling into the deep dish pie crust.
Top with the second pie crust, and crimp the edges of the crust together. Cut 4 slits for venting.
Beat the egg with the milk, and brush the egg wash over the top crust.
Place the pie plate on a cookie sheet, to catch any drips, and bake at 375° F for about 45 minutes, or 'til the crust is golden. Tent the edges with foil, or use a pie shield to protect the edges from burning.
When the pie is done, remove it from the oven, and allow it to cool for 5-10 minutes before serving.