Preheat the oven to 375°F.
In a large skillet over medium-low heat melt the butter and add the onion, carrot, and celery. Cook until softened and the onions are translucent, about 10 minutes stirring occasionally.
Add in the garlic, thyme, and rosemary, cook for 30 seconds stirring constantly until fragrant.
Add the flour to the pan and whisk it in to combine, cook for 2 minutes, stirring occasionally.
Slowly add in the half and half and the stock while whisking constantly to avoid lumps. Bring to a simmer, and simmer until thickened, about 5 minutes.
Take off the heat, stir in the turkey, salt, pepper, and the peas.
Grease a 10-inch baking dish and pour the filling into the dish.
Top with the pie crust, add a few slits in the top for air holes.
In a small bowl whisk the egg with a splash of water. Brush the egg wash on top of the pie crust.
Bake for 30-35 minutes until golden brown.
Let sit for about 10 minutes, cut and serve.