Preheat the oven to 350 degrees and line an 8x8 baking pan with parchment paper. Spray with nonstick cooking spray and set aside.
In a medium bowl, whisk together the flour, cinnamon, baking soda, baking powder, salt, ginger, and nutmeg. Set aside.
In a separate bowl, beat together the butter, sugar, and brown sugar until light and fluffy.
Add the eggs and vanilla and beat until smooth.
Add half the dry ingredients and beat until mixed, then add the remaining dry ingredients and beat until smooth.
Fold in the shredded carrots.
Spread the batter in the prepared baking pan and bake for 35 to 40 minutes, until a toothpick inserted into the cake comes out clean.
Allow the cake bars to cool completely, then prepare the icing by beating together the cream cheese, butter, and vanilla. Add the powdered sugar one cup at a time, beating between each addition.
Separate a small amount of the icing into two separate bowls. Add the orange food coloring to one bowl and the green food coloring to the other bowl. Stir until the coloring is mixed into the icing, then transfer each color to a piping bag.
Remove the cake bars from the pan by pulling the edges of the parchment paper up. Spread the remaining icing over the top of the cake bars and cut the bars into 12 pieces.
Decorate the top of each cake bar with a carrot using the orange and green icing.