Preheat the oven to 350F, and spray a 9 inch baking dish with nonstick cooking spray.
Using a food processor shred the carrots, then pat them dry with a paper towel.
In a large bowl, combine the egg, egg white, nonfat Greek yogurt, and milk. Mix well and then stir in the oil and vanilla extract. Mix in the sugar, all purpose flour, spices, baking powder, baking soda, and mix well.
Fold in the carrots and nuts.
Pour the batter out into the baking dish, using a spatula to get all the batter into the dish.
Bake the bars in the oven for 40 minutes. Let them cool completely. The cake will settle once it is out of the oven.
Mix together all of the icing ingredients until its light and fluffy. Let it chill in the fridge for at least 30 minutes.
Once the bars are cooled, cut them into 9 and top with the chilled icing. Enjoy!