Preheat oven to 350°F and grease a 9x13 inch baking pan.
In a mixing bowl, stir together the flour, baking powder, baking soda, salt and cinnamon; set aside.
2 cups flour, 2 teaspoons baking powder, 1 teaspoon baking soda, 1/2 teaspoon salt, 2 teaspoons cinnamon
In a stand mixer or large mixing bowl, combine the oil, sugar, brown sugar, eggs, applesauce and vanilla; add the shredded carrots and mix well.
1 cup vegetable oil, 1 cup sugar, 1 cup brown sugar, 4 eggs, 1/2 cup applesauce, 1 teaspoon vanilla, 3 cups shredded carrots
Add the dry ingredients to the wet ingredients and mix until just combined (don't overmix).
Pour into prepared sheet pan and spread evenly.
Bake for 30-35 minutes, or until a toothpick in the center comes out clean. (A few moist crumbs are fine, but you don't want it to be gooey). Let cool completely.
Make the frosting: Beat the cream cheese and butter until fluffy. Beat in the powdered sugar and vanilla until smooth.
1 8-ounce package cream cheese, 3 tablespoons butter, 3 cups powdered sugar, 1 teaspoon vanilla
Spread the frosting over the cooled carrot cake. Decorate with sprinkles, pecans, or little candy carrots.
Store the finished cake in the refrigerator.