Preheat oven to 375 degrees. Grease a loaf pan and set aside.
In a small mixing bowl add the eggs, vanilla extract, melted butter and applesauce. Stir until completely combined. Mix in the sugars. Stir until combined.
In a large mixing bowl add the flour, salt, baking powder, baking soda, cinnamon, and nutmeg. Stir with a fork to combine. Pour the wet ingredients into the dry ingredients and stir to combine. Add in the shredded apple and stir.
Pour dough into prepared loaf pan. Bake at 375 degrees for 50-60 minutes, top will be golden brown. Check that it's cooked by inserting a toothpick in the center, once it's done cooking the toothpick will come out clean, no dough will be stuck on it.
Allow bread to cool in pan for about 5-10 minutes. Turn upside down and remove bread loaf from pan and allow to continue cooling on wire rack.
Store covered at room temperature for 3-4 days.