Preheat oven to 400°F.
Pour oil into a medium skillet. Place over medium heat.
When oil is hot and shimmering, add prepared onion, carrots and celery. Sauté until browned.
Add sliced mushrooms. Sauté until browned and caramelized.
Add sliced mushrooms. Sauté until browned and caramelized, about 4 minutes.
Add diced turkey, gravy and thyme. Bring filling mixture to simmer.
Taste. Season, to taste, with kosher salt and freshly ground pepper.
Mix in peas and chopped parsley.
Pour filling into 9-inch deep dish pie tin.
Lay out prepared top pie crust and cut out vent holes. (Simple slashes or decorations like leaves). Place crust on top filling and pinch edges of crust all around.
Brush crust with egg wash.
Place in preheated oven and bake until crust is golden and filling is bubbling, 30 to 35 minutes.
Remove and let cool slightly before serving. Serve with cranberry sauce.