Preheat the oven to 325 degrees F. Generously spray the inside of a 10-inch bundt pan with nonstick cooking spray or butter it, dust with flour, and knock out the excess flour.
In a small saucepan, heat the Dr. Pepper, 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 to 40 minutes, rotating the pan halfway through the baking time, until a toothpick 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.
2 ounces dark chocolate, melted and cooled slightly
1/2 cup (1 stick) unsalted butter, softened
1 teaspoons salt
1/4 cup root Dr. Pepper
2/3 cup dark unsweetened cocoa powder
2 1/2 cups powdered sugar
Beat all ingredients together with an electric mixer until a good consistency.