Preheat your oven to 350°F (175°C). Grease a 9 x 13-inch baking pan with nonstick spray or a thin layer of oil to prevent the cake from sticking.
Using the coarse side of a grater, grate two small to medium zucchinis over a clean tea towel. Fold the towel around the shredded zucchini and twist tightly over the sink to squeeze out as much excess liquid as possible. Measure out 2 cups of the drained shredded zucchini and set aside.
In a small mixing bowl, whisk together the flour, baking powder, baking soda, salt, and ground cinnamon until they’re well combined. Set aside for later use.
In a large mixing bowl, beat the eggs until well combined, then whisk in the white sugar, vanilla extract, and oil. Once the wet ingredients are smooth, add the dry ingredients from Step 3 and mix until a thick batter forms. Finally, fold in the drained shredded zucchini from Step 2 and stir until just combined. I find that mixing by hand at this stage keeps the cake tender.
Pour the batter into the greased 9 x 13-inch pan from Step 1. Spread it evenly with a spatula or spoon. Place the pan on the bottom rack of your preheated oven and bake for about 35 minutes, or until the surface is golden and a toothpick inserted in the center comes out clean. Let the cake cool completely in the pan, about 1 hour, before adding any icing.