Preheat oven to 350-degrees. Line a 9 x 13 pan and set aside.
In a large mixing bowl, whisk together the flour, baking soda, baking powder, cinnamon, ginger, cloves, and salt.
In a separate mixing bowl, whisk the eggs. Slowly whisk in the slightly cooled melted butter. Add the granulated sugar, brown sugar, and vanilla extract, and mix until smooth and well combined.
Grate two large zucchinis.
Mix in the wet ingredients into the dry ingredient bowl, fold in the grated zucchini. Pour the wet ingredients into the dry ingredients and mix until just combined. Gently fold in the grated zucchini.
Pour the batter into the prepared baking dish and spread evenly. Bake for about 45 minutes, or until a toothpick inserted into the center comes out clean. Be careful not to overbake to keep the bars soft and moist.
Fluffy Vanilla Buttercream
Using an electric mixer, beat the softened butter until light and fluffy. Gradually add the powdered sugar, mixing until smooth. Add the vanilla extract and milk, then beat on high until the buttercream is light, fluffy, and creamy. Spread over the completely cooled zucchini cake bars.