Preheat your oven to 350°F (175°C) and grease a 9-inch round cake pan with butter or non-stick spray.

In a medium bowl, whisk together the dry ingredients: flour, baking soda, baking powder, cinnamon, nutmeg, and salt. Set aside.

In a large mixing bowl, beat the softened butter, granulated sugar, and brown sugar until light and fluffy.

Add the eggs one at a time to the butter-sugar mixture, mixing well after each addition. Follow with the vanilla extract.

Gradually add the dry mixture into the wet ingredients, alternating with the applesauce. Fold in the diced apples.

Pour the batter into the prepared cake pan and bake for about 40 minutes, or until a toothpick comes out clean.

Allow the cake to cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.

For the icing, combine butter, brown sugar, and milk in a saucepan. Bring to a boil, stirring constantly.

Remove from heat, whisk in the powdered sugar until smooth. Adjust with more milk if needed.

Once the cake has cooled, pour the icing over the top, allowing it to drip down the sides.