Preheat oven to 350°. Spray two 9" cake pans with nonstick spray and line the bottoms with parchment paper if you like. It makes them come out of the pan very easily so I would recommend! Set aside.
In a large bowl, mix the flour, baking powder, baking soda, cinnamon, ginger, nutmeg and salt. Set aside. In a separate bowl, whisk together the oil, eggs, brown sugar, white sugar, applesauce and vanilla. Once combined, fold your shredded carrots into the wet mixture. Now add your wet ingredients to your dry and fold until combined. Don't over mix but make sure you don't see any clumps of flour anywhere.
Divide the batter evenly into your prepared pans and bake for 30-35 minutes until toothpick comes out clean. Mine ware perfect right around 33 minutes. Take out of oven and let cool in pans for 15 minutes. Then flip onto a cooling rack and cool for another 25 minutes.
While the cakes are cooling, make the frosting. In a medium bowl using a hand mixer, beat your cream cheese and butter together until combined and slightly fluffy. About a minute. Then add in the powdered sugar, vanilla and lemon juice. Mix until all combined, scraping down the sides as necessary.
Once cakes are cooled, frost your cake. Place one layer down on a cake plate and spoon over a few spoonfuls (about 1/2 cup-3/4 cup) and spread on the top only. Then place your second layer on top of that and continue frosting to your liking. I topped my cake with some chopped walnuts.