Preheat the oven to 350 degrees F (175 degrees C). Line a baking sheet with parchment paper. Lightly butter a 2-quart soufflé or ceramic baking dish; sprinkle 1 tablespoon white sugar over butter, tapping excess sugar out of the dish. Store the prepared dish in the refrigerator.
Place butternut squash, cut-sides 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 together flour, baking powder, and salt in a bowl; set aside.
Scrape flesh from butternut squash into a food processor and process until smooth. Add flour mixture, butter, 1/4 cup white sugar, brown sugar, vanilla, cinnamon, and nutmeg; process until smooth. Add egg yolks, one at a time, to squash mixture while continuously processing. Transfer squash mixture to a large bowl.
Beat egg whites in a bowl with an electric mixer until stiff peaks form; fold into squash mixture. Pour mixture into the chilled dish.
Bake in the preheated oven until the top is browned and springs back when gently pressed, about 1 hour.