First, preheat your oven to 350 degrees F. Line an 8x8 baking pan with parchment paper and/or spray it.
To a large bowl add the eggs, applesauce, oil, vanilla extract and maple syrup. Whisk well.
Next, add in the flours, pumpkin pie spice, salt, baking powder and baking soda. Use a rubber spatula to combine.
When the flour is almost all mixed in, add the carrots and walnuts and combine just until incorporated.
Spoon the mixture into the prepared 8×8 baking pan. Bake at 350 degrees F for about 30-35 minutes, or until a toothpick inserted comes out clean.
Once baked, let the cake cool for about 8 minutes in the baking pan. Then, transfer it to a cooling rack to cool completely. Make the frosting as the cake is cooling.
Use a mixer or food processor to make the frosting. Combine the yogurt, cream cheese and maple syrup until creamy. There should be no lumps.
Once the cake is cooled completely, evenly spread the frosting over the cake. I added a sprinkle of extra chopped walnuts over the top. Slice into 16 bars and enjoy!