Preheat the oven to 350ºF (177ºC). Grease and flour (or use homemade cake release) a bundt pan. Set aside.
In a medium bowl, whisk together the flour, baking soda, cinnamon, ginger, nutmeg, salt, and black pepper. Set aside.
In a large bowl, whisk together the melted butter, pumpkin purée, granulated sugar, brown sugar, eggs, and buttermilk until smooth. Gently stir in the flour mixture until everything is just combined. Pour the batter evenly into the prepared pans.
Bake the cake for 32-36 minutes or until a toothpick inserted into the cake comes out clean. Remove from oven and allow to cool for 15 minutes on a wire cooling rack.
Using oven mitts, carefully flip the bundt cake onto your serving dish/platter. To do this: flip the serving dish/platter over and place on top of the cooling bundt pan. Flip everything over and allow cake to cool completely in the pan, sitting on the serving dish. When cake is completely cool, lift pan off of the cake. Note: while the cake cools, brown the butter for the buttercream, as it will need to be refrigerated before use.
BROWN BUTTER BUTTERCREAM
In a heavy-bottomed large saucepan, melt the butter over medium heat. Stir frequently with a spatula or swirl the pan to cook the butter evenly. It will turn lemon yellow, then golden brown, then deep amber. The milk proteins will separate out and start to brown faster than the butter. This is ok. When the butter is deep amber and smells nutty, remove from heat and pour into a small bowl. Allow to cool about 15 minutes before chilling in the refrigerator until firm, about 1 hour.
In a large bowl with a handheld or the bowl of a stand mixer fitted with a paddle attachment, cream the butter on high speed until smooth and creamy. It may seem very loose/melted, but this is ok.
With the mixer on low, add the powdered sugar, vanilla extract, and salt. Increase mixer speed to medium-high and beat for 2 minutes.
Test the consistency of the buttercream and add milk or cream little by little to achieve desired consistency.
Spread brown butter buttercream over the cake using a spatula and serve. Cake stays fresh covered in the refrigerator for up to 7 days. Cake freezes well, up to 2 months. Wrap in plastic wrap, then in foil, then place in a large freezer bag or freezer-safe container. Thaw in refrigerator overnight.
Room temperature eggs: it is important to use room temperature eggs when working with room temperature butter as it incorporates into batter much more evenly.
Buttermilk: if you don't have buttermilk, you can make your own: see my post about homemade buttermilk.