Preheat the oven to 350°F and spray two 9-inch cake pans with non-stick spray and line the bottoms with circles of parchment paper.
Sift the all purpose flour and cocoa powder into a mixing bowl and add the white and brown sugar, baking soda, baking powder, espresso powder and salt. Mix together.
Add shredded zucchini, oil, eggs, Greek yogurt and vanilla extract.
Mix into a thick batter. If it doesn't want to come together, leave the bowl for a minute and then come back and mix again. The zucchini will release extra moisture into the batter and make it easier to mix.
Transfer the batter to your prepared 9-inch baking pans and smooth down.
Bake for 30 minutes and then do the toothpick test - insert a toothpick into the center of one of the cakes and it should come out clean.
Allow the cakes to cool in the pans for a few minutes and then transfer them to a wire cooling rack to cool completely before frosting.