Make pastry as directed in a 9-in. deep-dish pie pan, rolling dough to 13 in. and baking it blind; let cool.
Preheat oven to 375° (skip this step if using canned pumpkin). Cut squash in half lengthwise, using a large, heavy knife. Scrape out seeds with a large spoon. Set squash cut side down on a rimmed baking sheet. Roast until very soft when pierced, 45 minutes to 1 hour. Let cool.
Meanwhile, finely grind gingersnaps in a food processor. Add butter and a pinch of salt and pulse until evenly moistened. Scrape into a bowl and set aside. Wipe crumbs from processor bowl and blade.
Trim browned edges from squash. Scoop flesh into food processor and purée until very smooth. If using any fresh pumpkin (besides kabocha) or canned pumpkin, simmer in a saucepan, stirring often, until very thick, 10 to 15 minutes. Measure 1 1/2 cups squash or pumpkin; save the rest for soup or another use.
Firmly and evenly press gingersnap mixture over bottom of pastry. Chill.
Preheat oven to 375°. In a small bowl, rub sugar and cornstarch with your fingers until no lumps remain. Add to food processor with the 1 1/2 cups squash, the crème fraîche, vanilla, 1/2 tsp. salt, the nutmeg, and eggs. Whirl until well combined, scraping inside of bowl. Pour mixture evenly into prepared crust. Wrap pie rim with 4 strips of foil. (Bake any leftover filling in a ramekin.)
Bake pie 15 minutes (35 minutes for a ceramic pie dish); remove foil. Continue baking until center jiggles only slightly when gently shaken, 30 to 35 minutes more (15 minutes for a ceramic dish). Let cool on a rack until room temperature, 2 to 3 hours. Serve with whipped cream.
*Whole cane sugar, such as Rapunzel brand, has a complex, caramelized flavor; find it at natural-foods markets or online. If you use regular sugar, the pie will be a bit sweeter.
Make ahead: Through step 5, up to 1 day; chill squash. Or the whole pie, up to 1 day at room temperature.