Preheat oven to 350°F. Spray a 12-cup bundt pan with nonstick cooking spray (the kind with flour in it).
Place the butter in a small sauce pan over medium-low heat. Melt, then stir often, cooking until the butter browns. This will take 5-7 minutes or so. Stir often. The butter will foam, then reduce. Once it starts to turn brown it burns quickly so be sure and watch it. Remove from heat.
Measure out 1/3 cup of the browned butter. Reserve the rest for the glaze. Place the 1/3 cup brown butter with the cake mix, pumpkin puree, eggs, granulated sugar, and cinnamon, nutmeg, ginger, and allspice in a large bowl. Mix with a hand mixer until smooth. The batter is thick.
Pour batter into prepared pan. Bake for 35-38 minutes until a toothpick comes out clean. Cool completely before removing from pan and glazing.
To make the glaze: Place reserved brown butter in a bowl. If it has solidified, heat it in the microwave for a few seconds until it becomes liquid again. Whisk in powdered sugar, vanilla, and salt. Add heavy whipping cream 1 tablespoon at a time until you get the desired consistency. (I used 4 tablespoons.) Drizzle over cake.
Store loosely covered on the counter or in the refrigerator for up to 3 days.