Preheat your oven to 350°F (175°C). Grease and flour an 8×8-inch baking dish.
In a medium bowl, whisk together the flour, baking soda, baking powder, cinnamon, nutmeg, cloves, and salt.
In a large mixing bowl, cream together the softened butter and granulated sugar until it is light and fluffy. Beat in the egg, then stir in the applesauce and vanilla extract until everything is combined well.
Gradually add the dry ingredients to the wet mixture, mixing until just combined. Pour the batter into the prepared baking dish and smooth the top.
Bake in the preheated oven for 30-35 minutes, or until a toothpick inserted into the center comes out clean. Allow the cake to cool completely before frosting.
In another large bowl, beat together softened cream cheese and butter until smooth. Gradually add powdered sugar, vanilla extract, and cinnamon. Beat until it is light, fluffy, and spreadable.
Spread the cinnamon cream cheese frosting evenly over the cooled cake. Slice into squares and serve.