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 8-10 minutes.
3/4 cup (168 g) unsalted butter
Transfer the browned butter to a large bowl, then pop it in the fridge and allow it to cool completely until it is solid but still malleable. While it's cooling in the fridge, stir it with a rubber spatula every 10 minutes to make sure it cools evenly.
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, baking powder, baking soda, and salt. Then set aside the flour mixture.
2 1/4 cups (282 g) all-purpose flour, spooned and leveled, 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 brown sugar to the whipped brown butter. Cream together for 2 minutes on high speed.
1 1/3 cups (294 g) light brown sugar, packed
Then add in the eggs and vanilla and mix on medium speed until pale in color and smooth, about 1 minute.
3 eggs, at room temperature, 1 tbsp vanilla
Add in the buttermilk and 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.
1 cup (240 ml) buttermilk, at room temperature
Pour the batter into the prepared baking pan. Bake the cake for 38-42 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 cooling rack until completely cooled.