Preheat the oven to 325ºF. Generously grease the inside of a 10-inch bundt pan with butter or shortening and dust it with flour. Tap out any excess flour.

Combine the root beer, cocoa powder, and butter in a saucepan over medium heat. Stir until the butter is melted. Add the granulated sugar and the brown sugar and whisk to dissolve. Remove from the heat and stir in the root beer extract.

In a large bowl, whisk together the flour, baking soda and salt.

Whisk the eggs in a small bowl, then whisk the eggs into the cocoa mixture. Add the cocoa mixture to the large bowl with the flour mixture and fold together. Do not overmix - the batter will be a little bit lumpy.

Pour the batter into the prepared pan and bake for 35 to 40 minutes, rotating the pan halfway through the bake time, until a tester inserted in the middle comes out clean. Transfer to a wire rack to cool completely. Gently turn the cake out onto the rack.

To make the frosting, combine the melted and cooled chocolate, the butter, and the salt in a large bowl and beat to combine. Slowly beat in the root beer and the root beer extract, followed by the cocoa powder. Add the powdered sugar in 1/2 cup at a time until it is all combined.  Add heavy cream, if needed, until you have the correct consistency. The frosting will still be thick, but creamy enough to spread on top of the cake.

To assemble, spread the frosting on top of the cooled cake in a thick layer.