Preheat oven to 350 degrees. Spray 2 8-inch pans with non-stick baking spray. Dust both pans with flour and tap out any excess. This is necessary so the cake doesn't stick! Baking spray with flour works great as well.
In a large bowl, sift in pumpkin pie spice, cinnamon, cloves, ginger, baking soda, baking powder, and flour. Whisk until combined.
In a separate mixing bowl, add in the butter, oil, brown sugar, and granulated sugar. Mix on medium until light and fluffy. Scrape down the sides of the bowl.
Add in the vanilla and the eggs one at a time and beat until completely incorporated. Scrape down the sides of the bowl after each egg.
Add in the pumpkin puree and mix until incorporated.
Add in the dry ingredients and mix on low just until the flour is worked into the batter. Don't over mix!
Divide the cakes equally into the 2 8-inch pans.
Place into the oven for 30 - 35 minutes or until done. Place a toothpick into the center of the cake and if it comes out dry or with a few moist crumbs, it's done!
Let cool in the pan for 10 minutes and then turn out onto a cooling rack to cool completely.
In a saucepan over medium high heat, add in the butter. Let it melt and get a deep brown color. It still start to smell nutty. I cooked mine for 10 minutes.
As soon as the butter is brown, turn off the heat and without stirring at all, pour into a heat-safe dish. You can leave out any of the dark bits on the bottom of the pan, but I added them to the frosting for a more nutty flavor.
Let it cool down for 15 - 20 minutes in the fridge.
Add in the vanilla, powdered sugar, and milk. Mix on low until the powdered sugar is mostly incorporated and then whip for 1 minute on high.
Frost the cooled cake. Cut and serve!
When I doubled the brown butter frosting recipe, I still only used 1 vanilla bean. You can use 2 tsp. vanilla bean paste instead.