Preheat the oven to 375 degrees. Generously spray the inside of a 10-inch bundt pan with nonstick cooking spray; alternatively butter it, dust it with flour, and knock out the excess flour.
In a small saucepan, heat the root beer, cocoa powder, and butter over medium heat until the butter is melted. Add the sugars and whisk until dissolved. Remove from the heat and let cool.
In a large bowl, whisk the flour, baking soda, and salt together.
In a small bowl, whisk the eggs until just beaten, then whisk them into the cooled cocoa mixture until combined. Gently fold the flour mixture into the cocoa mixture. The batter will be slightly lumpy -- do not overbeat, as it could cause the cake to be tough.
Pour the batter into the prepared pan and bake for 35-40 minutes, rotating the pan halfway through the baking time, until a small sharp knife inserted into the cake comes out clean. Transfer the pan to a wire rack to cool completely. Gently loosen the sides of the cake from the pan and turn it out onto the rack.
To make the frosting, pulse all ingredients in a food precessor until smooth and shiny. I don't have a food processor, so I just made it like regular frosting. I cut the butter into the dry ingredients and then stirred in the root beer. Use a spatula to spread the frosting over the crown of the bundt in a thick layer and allow to set.