Preheat oven to 350℉. Line an 8x8 inch brownie pan with parchment paper.
In a large bowl, add in the eggs, applesauce, olive oil, vanilla and whisk together until combined. Next, add in the flour, baking powder, salt, cinnamon, nutmeg, ginger, all spice, sugar and mix together until incorporated. Fold in the apple pieces. Then, transfer mixture to the pan and spread out evenly. Bake for 30-35 minutes. Allow to cool completely before adding the icing. Poke some holes halfway through the cake. Add the icing on top.
Cinnamon Icing:
In a small bowl, add in the powdered sugar, milk, cinnamon and mix until combined. Taste and adjust if needed.