1. Preheat oven to 325 degrees F*. Generously spray the inside of a 10-inch bundt pan with nonstick cooking spray, or butter the pan and dust with flour, shaking out the excess flour; set aside.
2. In a medium 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.
3. In a medium bowl, whisk the flour, baking soda and salt together.
4. 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, which is okay. Do not overbeat it, as it could cause the cake to be tough.
5. Pour the batter into the prepared pan and bake for 35 to 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.
6. To make the Root Beer Fudge Frosting, put all of the ingredients in a food processor. Pulse in short bursts until the frosting is shiny and satiny, scraping the sides of the food processor a couple of times. (If you don't have a food processor, simply throw it all into the bowl of a stand mixer or a large bowl using a hand mixer and mix on medium-low until combined and satiny smooth.)
7. Use a spatula to spread the fudge frosting over the cake in a thick layer. Let the frosting set before serving. Store leftovers wrapped well or in an airtight container at room temperature.
*Note: If you are using a dark, nonstick pan, heat the oven to 300 degrees F.