Preheat the oven to 350 F, and position a rack in the center of the oven. Thoroughly grease all the crevices of an original 12-cup bundt pan with non-stick baking spray. If you have two of these pans, you can bake both halves of the cake at once, but if you only have one pan, you'll have to bake half the batter at a time. Place the butter in a saucepan over medium low heat. Melt the butter, and then continue to cook, swirling occasionally, until nutty brown solids form on the bottom of the pan. Set aside to cool for about 10 minutes.
In a large bowl, whisk together the browned butter with the pumpkin, brown sugar, eggs, milk and vanilla until smooth.
In a separate bowl, combine the flour, baking soda, salt, cinnamon, ginger, cloves and nutmeg. Add the dry ingredients to the wet and stir just until moistened.
Pour half the batter into the prepared pan, smoothing out the batter with a spatula. Cover the bowl of leftover batter, and leave it to rest at room temperature until needed.
Bake the cake for about 27-30 minutes, until a toothpick inserted comes out clean or with moist crumbs clinging to it.
Set the pan on a cooling rack, let cool for 15 minutes, then turn the cake out of the pan onto the cooling rack.
Spray the pan again with non-stick baking spray, and fill with the rest of the batter (do not stir the batter again, just gently scrape it into the pan). Bake the second cake, cool in the pan for 15 minutes, then turn the cake out onto the cooling rack.
Let the cakes cool completely before assembling and frosting. The cakes can be baked one day in advance, wrapped in plastic wrap while slightly warm, and kept at room temperature until you're ready to assemble the cake.