Preheat the oven to 350 F. Grease a 9x13 inch pan and flour it.
To a large bowl add eggs, oil, white sugar, and 2 teaspoons vanilla and beat it.
Add flour, baking soda, baking powder, salt, cinnamon, carrots, and pecans.
Transfer the mixture to the prepared pan.
Bake in the preheated oven for 40 to 50 minutes. Once baked, let it cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.
In a bowl beat applesauce, vegan cream cheese, confectioners' sugar, and 1 teaspoon vanilla until the mixture is smooth.
Add chopped pecans and top the cake with frosting.