Preheat the oven to 350 degrees F. Spray a 9x13-inch baking dish with non-stick baking spray, set aside.
In a medium-sized bowl, stir together the flour, baking soda, cinnamon, salt, cloves, and nutmeg, set aside.
In a large bowl, whisk together the sugar, applesauce, eggs, and vanilla until smooth.
Add the dry ingredients to the wet ingredients and stir it together until just combined, with no dry patches.
Add the apples and fold them in.
Pour the cake batter into the prepared baking dish and smooth out the top.
Place in the oven and bake for 30-35 minutes until browned and a toothpick inserted into the center comes out clean.
Remove from the oven and place the baking dish on a wire rack and let it cool completely.
While the cake is cooling, make the frosting by creaming the butter in a large bowl with an electric hand mixer until smooth.
Add the powdered sugar a little at a time until fully combined.
Add the cinnamon, salt, milk, and vanilla and slowly stir it in.
Turn the mixer up to medium-high speed and whip the frosting until light and fluffy for 2 minutes.
Once the cake is cooled, add the frosting to the top of the cake and frost until the whole cake is covered.
Slice and serve.