Preheat oven to 350˚F and grease 3, 6-inch round cake pans with butter and line them with parchment paper on the bottom. Butter the parchment paper on top and bottom.
Stir the cinnamon, ginger, cardamom, all-spice, nutmeg, and pepper together in a small bowl. Reserve 1 teaspoon of the mixture for the frosting.
In the bowl of a stand mixer beat together the oil, sugar, vanilla, eggs, and Butternut Squash Puree until combined. Add in the flour, baking powder, baking soda, salt, and the spice mixture, minus 1 teaspoon. Stir on low for 1 minute until no flour lumps remain.
Pour the batter into the cake pans spreading it evenly between them.
Bake 30-40 minutes, until the tops are just set, and no longer wiggly in the center. Remove and let cool 10-15 minutes, then run a knife around the edges of the pan and turn the cakes out onto a cooling rack. Cover and let the cakes cool completely before assembling.
If your cakes have domed, trim the tops to make them level and even.
Add 2 sticks butter to a skillet, set over medium heat. Allow the butter to brown lightly until it smells toasted, about 2-3 minutes. Stir often.
Remove from the heat and transfer the butter to the mixing bowl, let cool until it has solidified at room temperature.
Add the remaining stick of room temperature butter and the cream cheese to the bowl.
Beat the cooled brown butter, butter and cream cheese together for 2 minutes until light and fluffy.
Add the powdered sugar and maple syrup to the bowl and continue beating together.
Add the vanilla and reserved spice mix and beat until well combined.
Place one cake layer on a serving plate.
Spread 1/3 of the buttercream over the cake. Repeat with the remaining 2 cake layers.
Frost the outside of the cake and chill 30 minutes up to overnight. Cake will keep in the fridge for up to 4 days.