Preheat oven to 350°F (175°C). Grease a 9×13-inch baking pan.
Whisk flour, baking soda, baking powder, cinnamon, nutmeg, and salt.
In another bowl, whisk eggs, sugars, oil, applesauce, and vanilla until smooth.
Fold dry ingredients into wet mixture. Add apples and nuts.
Pour batter into pan. Bake 35–40 minutes or until a toothpick comes out clean.
Cool completely before frosting.
Beat cream cheese and butter until smooth. Add vanilla and salt. Gradually mix in powdered sugar.
Spread frosting over cooled cake and refrigerate until ready to serve.