Preheat the oven to 350 F and position a rack in the center of the oven. Thoroughly grease a standard 12-cup bundt pan with non-stick baking spray.
In a saucepan, melt the butter over medium heat. Continue to cook the butter, swirling the pan occasionally, until the surface of the butter begins to foam and nutty, golden brown solids have formed at the bottom of the pan.Remove the pan from the heat, and scrape the butter and browned bits into a large mixing bowl. Let cool for 5 minutes.
To the bowl of browned butter, add the pureed sweet potato, granulated sugar, brown sugar, eggs, milk and vanilla extract, and whisk until smooth.
In a separate bowl, sift together the flour, baking soda, salt and spices. Add the dry ingredients to the wet ingredients, and whisk until just combined. Pour the batter into the prepared pan.
Bake the cake for about 45 minutes. The top will have a deep, golden-brown crust with a cracked surface. When the cake is done, a cake tester or sharp knife inserted should come out clean, or with moist crumbs clinging to it.
Cool the cake in the pan for exactly 15 minutes, then turn the cake out onto a cooling rack. Cool the cake completely before frosting.