For the cake: Preheat the oven to 350 degrees F. Grease an 8-inch round pan or skillet with butter and lightly coat with flour.
In a medium bowl, whisk together the flour, crystallized ginger, baking soda, salt, ground ginger, cinnamon and cloves. Set aside.
In a large bowl, whisk together the butter, brown sugar, granulated sugar, molasses and milk. Whisk in the egg and orange zest.
Add the dry ingredients to the wet ingredients and whisk until just combined and there are no more lumps. Pour the batter into the prepared pan and bake for about 25 to 35 minutes, or until the edges look dark and the middle feels firm to the touch. Set the pan on a rack to cool.
For the eggnog royal icing: Measure out half of the powdered sugar and place in the bowl of a stand mixer or mixing bowl, then add the egg white, rum, bourbon, vanilla, nutmeg, cloves, salt, cream of tartar and heavy cream. Mix at low speed with a paddle attachment or with a hand mixer, adding the remaining powdered sugar. Increase speed to medium and beat until thick and smooth. It should have the texture of a loose buttercream, thick and malleable.
Pour the icing over the warm cake and spread evenly with a small spatula. Slice and serve.