Preheat oven to 350°F (175°C). Grease two 9-inch round cake pans, line bottoms with parchment paper, grease again, and lightly flour.
Peel and grate 4-5 medium carrots using the medium holes of a box grater to yield 3 cups. Set aside.
In a large bowl, whisk together flour, baking powder, baking soda, salt, cinnamon, nutmeg, and ginger.
In another large bowl, whisk eggs briefly, add both sugars and whisk 1-2 minutes until lighter. Add oil, applesauce, and vanilla; whisk until smooth.
Fold dry ingredients into wet mixture just until no flour streaks remain. Do not overmix.
Gently fold in the grated carrots until evenly distributed throughout the thick batter.
Divide batter evenly between prepared pans, smooth tops, and tap pans gently to release air bubbles. Bake 35-40 minutes until toothpick comes out clean.
Cool cakes in pans on wire racks for 10 minutes, then invert onto racks, peel off parchment, and cool completely to room temperature.
Beat softened cream cheese and butter until smooth and fluffy. Add vanilla and salt. Gradually add sifted powdered sugar on low, then beat on medium-high until fluffy.
Place one cooled layer on a serving plate. Spread 1-1 1/4 cups frosting on top. Add second layer, then frost top and sides. Refrigerate 30 minutes to set.