Preheat your oven to 350°F (175°C) and line an 8x8 inch baking pan with parchment paper.
In a medium bowl, whisk together flour, baking soda, baking powder, salt, cinnamon, nutmeg, and ginger.
In a large bowl, beat the eggs with granulated sugar and brown sugar until light and fluffy.
Add the oil, applesauce, and vanilla extract to the egg mixture and mix until combined.
Stir in the dry ingredients until just combined, then fold in the grated carrots.
Pour the batter into the prepared pan and smooth the top.
Bake for 25–30 minutes, or until a toothpick inserted into the center comes out clean. Let cool completely.
To make the frosting, beat the softened cream cheese and butter until smooth.
Gradually add powdered sugar and beat until fluffy. Mix in vanilla extract.
Spread the cream cheese frosting over the cooled carrot cake bars, slice, and serve!