Start by browning the butter. Add the butter to a medium pot and heat it over medium heat. Allow the butter to melt and come to a simmer. Simmer until the butter is foamy, giving off a nutty aroma and has turned a deep golden brown. The whole process should take about 10 minutes.
3/4 cup (168 g) unsalted butter
Transfer the brown butter to a large bowl. Allow it to cool in the fridge until the brown butter is about the same consistency as regular softened butter. Stir it every 10 minutes or so to ensure even cooling.
Preheat the oven to 350°F (175 ℃). Grease a 9×9 inch light metal baking pan and line it with parchment paper.
In a medium bowl, whisk together the flour, pumpkin pie spice, baking powder, baking soda, and salt. Then set aside the flour mixture.
2 1/4 cups (281 g) all purpose flour, spooned and leveled, 1 tbsp pumpkin pie spice, 1 1/2 tsp baking powder, 1/4 tsp baking soda, 1/2 tsp salt
Whip the cooled brown butter with an electric mixer on high speed until pale and fluffy. (If using a stand mixer, use a paddle attachment.) Add the sugar to the whipped brown butter. Cream together for 2 minutes on high speed.
1 1/2 cups (300 g) granulated white sugar
Then add in the eggs and vanilla and mix on medium speed until pale in color and smooth, about 1 minute.
2 eggs, at room temperature, 1 tbsp vanilla
Add in the buttermilk and pumpkin puree. Mix on medium-low speed until combined. (The batter will look split at this point, but don't worry. It will pull together once the dry ingredients are added in.) 1/2 cup (120 ml) buttermilk, at room temperature, 1 cup (244 g) canned pumpkin puree, at room temperature
Add in the dry ingredients, mixing on low speed until all is combined and the batter is smooth. Scrape the sides of the bowl with a rubber spatula as necessary.
Pour the batter into the prepared baking pan. Bake the cake for 35-38 minutes, or until a toothpick comes out clean from the center.
Let the cake cool in the pan on a wire rack for 30 minutes, then pull up on the parchment paper to release it from the pan. Let the cake continue cooling on the wire rack until completely cooled.