Preheat the oven to 350 degrees F (180 C). Coat the bottom and sides of an 8 or 9-inch square or round cake pan with butter, then lightly dust with flour. Set aside.

In a medium bowl, combine the flour, cocoa powder, baking soda, baking powder, and salt. Whisk briefly to combine, and set aside.

In a large bowl, use a hand or stand mixer to beat the butter, brown sugar, and granulated sugar until light and fluffy. Scrape down the bowl, then add the egg, egg yolk, and vanilla. Beat until well-combined, then add the buttermilk and beat again. Scrape down the bowl once more.

Add the dry ingredients to the egg-sugar mixture, and mix on low just until combined. Avoid overmixing - a few streaks of flour are OK! Scrape the batter into the prepared pan and smooth with the back of a spatula.

Bake for 25-30 minutes, until a toothpick inserted into the center comes out clean or with moist crumbs. Let cool for 5-10 minutes in the pan, then flip the cake out onto a cooling rack to cool completely before frosting.

Prepare the frosting by placing the butter in a small bowl. If it is not yet very soft, microwave on medium power for 10-20 seconds, then stir. It’s OK to have a few pools where the butter has melted. Add the cocoa powder and stir with a spatula to combine. Add the powdered sugar, vanilla, and milk, and stir until the frosting is even and thick. Add a pinch of additional powdered sugar or milk to reach your desired consistency. Frost and garnish cooled cake as desired.