Special equipment: two 9-inch cake pans
For the cake: Preheat the oven to 350 degrees F. Butter and flour two 9-inch cake pans.
Grate the zucchini on the large holes of a box grater, then wrap in a clean kitchen towel and squeeze dry.
Whisk the flour, baking powder, baking soda, salt, cinnamon, ground ginger and cloves together in a large bowl.
In a second large bowl, whisk the brown sugar, eggs, oil, milk, fresh ginger, vanilla and grated zucchini. Add the flour mixture to the wet ingredients and stir until just combined. Divide the batter between the prepared cake pans; smooth the tops with an offset spatula.
Bake the cakes until a wooden pick inserted into the center comes out clean, about 30 minutes. Cool in the pans for 10 minutes, then invert onto a cooling rack and cool completely.
Use a serrated knife to carefully cut each cake in half horizontally, making four layers.
For the frosting: Beat the cream cheese and butter in a large bowl with an electric mixer until smooth. Gradually add the confectioners' sugar, beating until incorporated. Beat in the chopped crystalized ginger.
To assemble the cake: Place a cake layer on a cake stand or platter. Spoon a quarter of the frosting (about 1 1/2 cups) on top and smooth it evenly with an offset spatula. Stack another cake layer on top; repeat with the frosting. Continue with the remaining cake layers and frosting, leaving the sides of the cake bare. Top with the sliced crystalized ginger.