Preheat oven to 350°F. Grease and flour two 8x4-inch loaf pans.
Whisk salt, baking powder, baking soda, cinnamon, ginger, nutmeg, and flour in a medium bowl. Set aside.
Whisk eggs until beaten, then whisk in oil, vanilla, applesauce and both sugars. Whisk until combined, then add dry ingredients and stir with a spatula or wooden spoon until just combined. Stir in carrots.
Pour batter evenly into the prepared pans. Bake for 40-55 minutes or until a toothpick comes out almost clean in the center.
Let the cake completely cool before removing from the pan and frosting.
To make the frosting: beat cream cheese and butter until smooth. Add powdered sugar, 1 cup at a time, until the mixture comes together, then add vanilla and salt. Beat until smooth
Remove cakes from pans and frost the top, then slice and serve.
Store loosely covered in the refrigerator for up to 3 days.