Preheat oven to 325 degrees. Butter a 9-inch Bundt pan. Dust with cocoa powder, and tap out excess; set aside. Put butter, molasses, brown sugar, and 1/4 cup water in a medium saucepan over medium-low heat. Cook, stirring constantly, until butter has melted. Transfer mixture to a large bowl. Let cool 5 minutes.
Add eggs, milk, and grated ginger to the molasses mixture; whisk to combine. Sift together flour, cocoa powder, baking soda, salt, ground ginger, and cinnamon into a medium bowl.
Gently fold the flour mixture into the molasses mixture until just combined. (There should be lumps remaining.) Pour batter into prepared pan. Bake cake until a cake tester inserted into center comes out clean, about 30 minutes. Let cake cool completely in pan on a wire rack.
Invert cake, and unmold onto a cake stand or a large serving platter. Using a spoon, drizzle the warm bourbon sauce over the cake in a back-and-forth motion. Serve immediately.