Add the butter to a large 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 10-15 minutes.

While it's still liquid, measure out 10 tablespoon of brown butter and add it to a large bowl, or the bowl of a stand mixer, to be used in the cake. Add the rest to a large, separate bowl to be used later in the frosting.

Let both portions cool in the fridge until solid but still soft.

Preheat the oven to 350 degrees. Grease a 9×9 inch 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.

Whip the smaller portion of brown butter with an electric mixer on high speed until pale and fluffy.

Add the granulated white sugar and brown sugar to the smaller portion of whipped brown butter. Cream together for 2 minutes.

Then add in the eggs and vanilla and mix on medium speed until pale in color and smooth, about 1 minute.

Add in the buttermilk and dry ingredients a little bit at a time, 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 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 cooling rack until completely cool.

While the cake bakes, make the easy caramel sauce. Add the butter and brown sugar to a medium saucepan. Bring to a simmer over medium heat and let simmer for 5 minutes, stirring occasionally.

Reduce to medium-low heat and whisk in the cream. Let the mixture simmer for 3-4 minutes.

Remove from the heat, then mix in the vanilla and salt. Pour the hot caramel into a bowl and let it cool on the counter to room temperature.

While the cake cools, make the brown butter cream cheese frosting by whipping the larger portion of brown butter on high speed with an electric mixer until pale and fluffy, 5-10 minutes.

Then add in the cold cream cheese and mix on medium speed until fluffy.

Sift in the powdered sugar and mix on low speed until combined, then on high speed for 1 minute until fluffy again.

When the cake is completely cooled, carefully pull up on the parchment paper to release the cake. Transfer it to a serving plate. Poke holes on the top of the cake using a wooden stick or the handle of a wooden spoon.

Transfer the cooled caramel sauce to a piping bag and cut a bit of the tip off. Pipe the caramel into the holes of the cake.