Preheat the oven to 350 degrees F. Generously butter and flour a 10 to 12-cup bundt pan.
Melt the butter in a large skillet over medium heat. Continue cooking until the foam in the pan has subsided and the butter is fragrant and nut brown, 3 to 5 minutes. Immediately transfer to the bowl of an electric mixer.
Whisk together the flour, baking powder, cinnamon, baking soda, allspice and salt in a large bowl.
Beat together the browned butter, sugars and eggs with an electric mixer until well combined and thick. Beat in 1 1/2 teaspoons of vanilla and the pumpkin. Fold in the flour mixture and the crystallized ginger.
Scrape into the pan and bake until a wooden pick inserted close to the center comes out with some moist crumbs attached, about 1 hour 15 minutes. Cool in the pan, then run a metal spatula around the edge and center of the pan and invert the cake onto a serving platter.
Stir together the confectioners' sugar, heavy cream and the remaining 1/2 teaspoon of vanilla in a small bowl. Drizzle over the cake.
While browning the butter for the cake makes it richer tasting, you can skip that step if you like. This is a full-flavored cake and one that keeps well. Store at room temperature for up to 3 days.