Make the crust: In a food processor or large bowl, combine flour and salt. Add cold butter and shortening; pulse or cut in until mixture resembles coarse crumbs with some pea-sized pieces of butter remaining (this creates flaky layers). Drizzle in ice water one tablespoon at a time, mixing gently until dough just comes together when pressed.
Chill the dough: Divide dough into two equal portions. Form each into a disc, wrap tightly in plastic wrap, and refrigerate for at least 1 hour (or up to 2 days).
Roset the squash: Preheat oven to 400°F. Toss squash pieces with 2 tablespoons olive oil, salt, and pepper on a parchment-lined baking sheet. Roast until fork-tender, about 35-40 minutes, turning once halfway through. Let cool completely.
Sauté the aromatics: While squash cools, melt remaining butter in a large skillet over medium heat. Add onion and cook until soft and translucent, about 8 minutes. Stir in garlic and cook for 30 seconds more.
Mash the filling: Transfer roasted squash to a food processor (or mash by hand). Process until smooth, then transfer to a large bowl. Whisk in eggs one at a time until fully incorporated. Stir in heavy cream, cinnamon, nutmeg, ginger, Gruyère, and Parmesan. Season generously with salt and pepper.
Add aromatics: Fold the sautéed onion mixture into the squash filling along with any accumulated juices from the pan (don't skip these — they add flavor).
Roll out crust: Roll one disc of dough on a lightly floured surface to about 12 inches in diameter. Line a 9-inch pie dish with it, pressing gently into the edges. Chill while you roll out the second disc.
Assemble: Pour filling into the crust-lined pan. Roll out top crust and place over filling; trim excess dough from edges and crimp to seal. Cut several slits in the top for steam to escape, or use a decorative cookie cutter if you're feeling fancy. Brush with an egg wash (1 beaten egg + 1 tsp water) and sprinkle with a pinch of salt.
Bake: Place pie on a baking sheet (to catch any drips). Bake at 375°F for 45-55 minutes, until the filling is set around the edges but still slightly jiggly in the center and the crust is golden brown.
Cool before slicing: Let pie rest on a wire rack for at least 20 minutes before cutting. This allows the filling to set properly so slices hold their shape.
