Preheat the oven to 375 degrees. Spray a jelly roll pan with cooking spray and place parchment on top.
In the bowl of a stand mixer, beat the eggs on medium speed until frothy and dark, about 5 minutes. Then add in the sugar, applesauce and vanilla extract.
In a medium bowl, whisk together the flour, baking powder, cinnamon, ground ginger, ground nutmeg and salt.
Add the grated carrots and flour mixture to the mixer and mix until just combined.
Pour the batter onto the prepared jelly roll pan. Use a spatula to spread the mixture into an even layer as it will be just enough to cover the entire pan.
Bake for 10-13 minutes or until the cake springs back when pressed with your finger.
Remove from the oven. Lift the parchment and cake off the pan and place on a heat-safe surface. Starting with the short end, roll up the cake and parchment together into a log. Place on a wire rack to cool completely.
While you wait, prepare the filling. Add the cream cheese, butter, powdered sugar, vanilla extract and salt to the bowl of your stand mixer and beat with the paddle attachment until light and smooth.
Once the cake has cooled, carefully unroll it and spread the cream cheese mixture all over the cake. Roll the cake back up again, gently peeling off the parchment paper as you go.
Wrap the cake in plastic wrap and refrigerate for one hour.
Slice a sliver off each end to remove the excess cake that may not have rolled up correctly. Transfer to a plate and dust with powdered sugar before slicing.