Prepare the Dough. In a large bowl, cream together the butter, granulated sugar, and brown sugar until light and fluffy. Add in the egg and vanilla extract, and mix until smooth.
Mix in the Applesauce. Gradually add in the applesauce and mix until well combined. The applesauce will make the dough soft and slightly sticky.
Combine Dry Ingredients. In a separate bowl, whisk together the flour, baking soda, cinnamon, nutmeg, and salt. Slowly add the dry ingredients to the wet ingredients, mixing until just combined. Be careful not to overmix, as this will keep the cookies soft.
Scoop and Bake. Using a cookie scoop, drop tablespoon-sized balls of dough onto a parchment-lined baking sheet. Flatten them slightly with the back of a spoon. Bake at 350°F for 10-12 minutes or until the edges are golden brown.
Cool and Enjoy. Allow the cookies to cool on the baking sheet for 5 minutes before transferring to a wire rack. Enjoy warm or at room temperature!