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Preheat the oven to 350 degrees. Grease an 8 or 9 inch round cake pan and line the bottom of the pan with a circle of parchment paper.
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In a large bowl, whisk together the flour, baking soda, baking powder, salt, cinnamon, and ginger. Add the shredded carrots and toss to coat. Make a well in the center of the dry ingredients.
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Add the sugars, canola oil, applesauce, vanilla extract, apple cider vinegar, and water. Stir well until combined.
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Pour the batter into the prepared pan. Bake at 350 degrees for 30-38 minutes (depending on the size of the pan), or until a toothpick inserted in the middle comes out clean.
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Let the cake cook for about 15 minutes in the pan, then run a knife around the edge of the pan and invert over a wire rack to remove. Let the cake cool fully on the wire rack.
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Make the frosting once the cake is fully cooled. In a mixing bowl whip the Daiya cream cheese and the vegan buttery spread. Sift in the powdered sugar. Mix until very smooth and creamy.
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Frost the cooled cake, and refrigerate if not serving immediately. Store leftover cake in the refrigerator.