Preheat oven to 350 degrees. Defrost your frozen GF pie shell atop a sheet pan while you begin sautéing your filling ingredients.
In a large saute pan over medium heat, add butter and olive oil and your butternut squash. Sauté until lightly browned and beginning to soften, about 5 minutes. Add your chicken sausage and mushrooms and sauté until sausage begins to brown and your butternut squash is tender, another 5 minutes. Stir in your minced shallot and fresh sage, saute another 5-7 minutes until shallots and mushrooms are golden and translucent. Set aside off the heat.
While vegetables are sauteing, whisk together your custard. In a large bowl, whisk together eggs, milk, spices and cheese until frothy and well combined. Let come to room temperature while par-baking crust.
With a fork, prick the bottom section of the pie crust all around. Bake at 350 degrees for 7 minutes, rotating once 180 degrees midway through cooking, to par-bake.
Stir in arugula to your sausage mixture then add mixture dispersed evenly in pie shell. Slowly pour custard evenly atop.
Bake until top is golden and eggs are just set- about 25-30 minutes, turning pan 180 degrees every 10 minutes throughout cooking. Let cool 10 minutes before slicing and serving.