To make the pumpkin brown butter cake, start by preheating the oven to 325 degrees F. Grease the bottom and sides of 2 8-inch round cake pans. Place round piece of parchment paper on the bottom of each cake pan and then grease the top of the parchment paper. Flour the sides and bottom of the cake pans, set aside until needed.
In a small saucepan, melt the butter over medium-low heat. Continue to cook until the butter has turned a golden brown color. There will be small black dots in the butter. Don’t worry this normally happens when you are browning butter.
Place a mesh strainer over a small bowl and pour the browned butter through the strainer. This will catch most of the burnt pieces in the butter. Set the bowl aside to cool.
In a medium bowl, whisk together the flour, baking soda, cinnamon, ginger, all spice, and salt. Set aside.
In the bowl of a stand mixer, beat together the sugar, brown sugar, pumpkin puree, eggs, and sour cream. With the mixer running on low speed, slowly add the flour mixture. Add the brown butter and mix until it is fully incorporated and the batter is smooth.
Divide the batter between the prepared cake pans. I use a scale to weigh the pans and make sure they are even. Smooth the tops of the batter and place the pans in the oven to bake. Bake until the tops turn a light golden brown and a toothpick inserted into the middle of each cake comes out clean, about 30-40 minutes.
Remove the cakes from the oven and let sit in the pans for 10 minutes. Carefully remove the cakes from the pans and remove the parchment paper from the bottom of each cake. Place the cakes on a cooling rack to cool completely.