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Adjust oven rack to the middle position, and preheat to 350 degrees F. Use cooking spray or butter to thoroughly grease the inside of either:
-1 13x19-inch baking pan, or
-2 9-inch round cake pans, or
-3 8-inch round cake pans
Then dust the inside of each pan with gluten-free flour until the grease is completely covered and no spots of the pan are visible.
In a separate bowl, whisk together the gluten-free flour blend, baking powder, baking soda, cinnamon, nutmeg, salt, cloves and ginger together until combined.
In a food processor fitted with the large shredding disk, shred carrots. Then transfer them to the mixing bowl with the flour. Wipe clean the food processor bowl, and fit with the metal blade (in place of the shredding disk). Add the granulated sugar, brown sugar and flax eggs and process until frothy and throughly combined, about 20 seconds. With processor running, add oil in a steady stream. Process until egg mixture is light in color and well emulsified, about 20 more seconds.
Transfer egg mixture to the large bowl with the carrots and flour. Using a rubber spatula, stir the mixture until thoroughly combined. If using nuts, stir them in until combined.
Pour batter evenly into the prepared pan(s). Bake until a toothpick inserted into the center of the cake comes out clean, about:
-30-35 minutes for the 9x13-inch baking pan
-25-30 minutes for the two 9-inch round cake pans, or
-18-23 minutes for the three 8-inch round cake pans
Let cake cool completely in the pan(s) on a wire rack, about 1 hour. Run a knife around the edge of the cake to loosen. Remove cake, if transferring to a different serving platter, and frost as desired.
Combine all ingredients in the bowl of a stand mixer or food processor, and mix until smooth and combined. If the frosting is too thick, add in some extra milk to thin. If the frosting is too thin, add in more powdered sugar to thicken.