Combine the eggs, oil, sugar and stir well. Make a well in the middle of this mixture and add the flour, soda, powder, salt and cinnamon. Stir just til combined. Fold in carrots. Bake in a greased 15x10 or 18x10 baking pan with rolled edges (cookie sheet with rim all the way around), at 350, for 25-35 minutes. GLUTEN FREE: If baking gluten free bake it at 325 for 35-45 minutes. Set cake aside to cool completely before frosting.
Using a handheld mixer beat the cream cheese, butter, powdered sugar and vanilla til smooth. VEGAN: If making vegan, optionally, chill the frosting for 30-60 minutes before frosting the cake to give it a little more structure. It is fine to frost immediately as well.
When baking gluten free it is best to go "slow and low", drop it 25 degrees from 350 to 325, and extend the baking time by about 7-10 minutes or until toothpick comes out clean from center.
For the cream cheese frosting when going vegan, to help with the consistency of the frosting, I made it straight from the fridge because it already has a softer texture due to a higher oil content.
For the cream cheese frosting whether making vegan or traditional, if it is too thick add a little milk or non-dairy alternative. If it is a little on the runny side, add a little more powdered sugar.