Preheat oven to 350 degrees F (175 degrees C). Line a baking sheet with parchment paper. Lightly butter a souffle or ceramic baking dish and sprinkle 1 tablespoon white sugar over butter, tapping excess sugar out of dish. Store buttered and sugared dish in the refrigerator.
Place butternut squash, cut-side down, onto the prepared baking sheet.
Bake in the preheated oven until very soft, 1 to 1 1/2 hours. Allow squash to cool to room temperature.
Sift flour, baking powder, and salt together in a bowl.
Scrape flesh from butternut squash into a in a food processor until smooth. Add flour mixture, 1/4 cup white sugar, brown sugar, vanilla extract, cinnamon, and nutmeg and process until smooth. Add egg yolks, 1 at a time, to squash mixture while continually processing.
Beat egg whites in a bowl using an electric mixer until stiff peaks form; fold into squash mixture. Pour mixture into the chilled dish.
Bake in the preheated oven until browned and top springs back after pressing, about 1 hour.