Preheat the oven to 350°F. Line cake pans with parchment.
Blend flour, salt, spices, baking powder, and soda in a bowl.
Whisk eggs, granulated sugar, and brown sugar until smooth.
Add oil, applesauce, and vanilla to the egg mixture.
Stir the dry and wet ingredients together gently.
Fold in grated carrots and walnuts with a spatula.
Split the batter equally between the two cake pans.
Bake for 30-35 minutes. Check with a toothpick.
Let the cakes rest on a rack for 15 minutes to cool. Remove from pans.
Beat cream cheese and butter until fluffy for the frosting.
Add powdered sugar, vanilla, and salt. Mix until smooth.
Spread frosting between layers and on top of the cake.
Chill the cake for 30 minutes before serving.