Preheat the oven to 350 degrees. Spray a 9x13 baking dish with cooking spray.
In a medium bowl, whisk together the eggs, applesauce, brown sugar, and vanilla until combined. Set aside.
In a large bowl, whisk together the flour, baking soda, cinnamon, baking powder and ground ginger until combined. Pour the wet ingredients into the dry and fold until just combined.
Add the grated carrots, pineapple and walnuts and fold until combined and no lumps of flour remain.
Pour into the prepared dish and bake for 50 minutes. Allow to cool completely on a wire rack.
While the cake is cooling, prepare the frosting. Add the cream cheese and butter to the bowl of a stand mixer and beat on medium speed until combined.
Add the brown sugar and beat until fully combined. Add in the vanilla and salt, then slowly add in the powdered sugar until fully combined.
Spread over the cooled cake and cut into slices.