Preheat oven to 350°F (180°C). Spray 3 (8-inch) round cake pans with baking spray with flour. Line bottom of pans with parchment paper.

In the bowl of a stand mixer fitted with the paddle attachment, beat butter and sugars at medium speed until fluffy, 2 to 3 minutes, stopping to scrape sides of bowl. Beat in pumpkin and vanilla bean paste.

In a medium bowl, whisk together flour, cornstarch, cinnamon, pie spice, baking soda, and salt. Add flour mixture to butter mixture all at once; beat at low speed until nearly combined. Fold in white chocolate until combined. Divide dough among prepared pans (a heaping 1½ cups or about 443 grams each), pressing into an even layer.

Bake until centers are golden brown and set and a wooden pick inserted in center comes out clean, 20 to 22 minutes, rotating pans halfway through baking. Let cool in pans for 10 minutes. Using a small offset spatula, loosen edges of cookie layers. Carefully invert onto wire racks; turn cookie layers parchment side down, and let cool completely.

Remove and discard parchment. Place 1 cooled cookie layer, flat side down, on a serving plate. Reserve 1½ cups (252 grams) Spiced German Buttercream. Spoon 1¾ cups (294 grams) remaining buttercream into a large pastry bag fitted with a ½-inch round piping tip (Wilton 1A); pipe evenly onto cookie layer, and smooth with a small offset spatula. Top with second cookie layer. Spoon 1¾ cups (294 grams) remaining buttercream into pastry bag, and pipe onto cookie layer, smoothing top. Place remaining cookie layer, flat side down, on top. Using a large offset spatula, spread remaining Spiced German Buttercream all over sides of cake and ½ to ¾ inch around edge of top. Using a bench scraper held flush against side of cake, scrape off excess buttercream to create a semi-naked cake look. (See Notes.)

Place reserved 1½ cups (252 grams) Spiced German Buttercream in a pastry bag fitted with a large French star piping tip (Ateco #868); pipe decoratively around top edge of cake as desired. Garnish with pumpkin seeds, if desired. Refrigerate for at least 25 minutes or up to overnight before slicing and serving. (See Notes.)