Preheat oven to 350°F (175°C). Line an 8x8 baking dish with parchment paper or spray with nonstick spray.
In a medium bowl, whisk together flour, cinnamon, baking soda, baking powder, salt, ginger, and nutmeg.
In a large bowl, beat softened butter with white and brown sugar until light and fluffy.
Add eggs one at a time, then mix in vanilla extract.
Gradually add dry ingredients to wet ingredients, mixing gently until just combined.
Fold in shredded carrots evenly.
Pour batter into prepared dish and bake for 35-40 minutes, or until a toothpick comes out clean.
Cool in pan for 10 minutes, then transfer to a wire rack to cool completely.
For frosting, beat cream cheese, butter, and vanilla until smooth.
Add powdered sugar one cup at a time, beating until creamy.
Spread frosting over cooled cake. If desired, tint small portions of frosting with food coloring for decoration.
Cut into 16 bars, serve, and enjoy!