Preheat oven to 350 degrees F. Grease bottom and 1/2-inch up sides and lightly flour two 8x4x2-inch foil loaf pans; set aside.
In a bowl, combine eggs, oil, applesauce, and sugar; beat with a whisk or rotary beater. Stir in apple and vanilla. In another bowl, combine flour, cinnamon, salt, baking powder, and baking soda; add to apple mixture. Stir to combine. Stir in nuts. Divide batter evenly between the prepared pans.
Bake for 50 to 60 minutes or until a wooden toothpick inserted near centers comes out clean. Cool in pans on a wire rack for 10 minutes. Remove from pans and cool completely. Wash and dry foil pans. Return cooled loaves to pans. Wrap tightly.