Grease a 9×9 cake pan with nonstick spray. Line with parchment and spray again. Then set aside.
In the bowl of an electric mixer, cream the butter and sugar until light and fluffy, 3 minutes.
Add the eggs one at a time, mixing in between each until incorporated.
Mix in applesauce. The batter will seem very loose and curdled, but that is completely normal.
Add the flour, baking powder, baking soda, cinnamon, cloves, nutmeg and salt. Mix until just combined, 2 minutes.
Remove bowl from mixture and use a spatula to scrape the bottom of bowl, ensuring the batter was mixed evenly.
Pour into prepared pan and bake until a toothpick or skewer inserted into the middle comes out clean, about 35-45 minutes. While the cake is cooling, make the frosting.
For the frosting
In a 4-quart saucepan, combine the butter, brown sugar and milk. Melt on medium low heat, stirring occasionally. Continue to cook until the mixture begins to boil and boil lightly for 2 minutes. Remove from heat and cool for 10-15 minutes.
Add vanilla and sugar. Whisk until smooth and thick.
Once the cake is completely cool, frost the cake. Sprinkle with toasted walnuts if desired.
To maintain freshness, this cake freezes well in an airtight container up to one month.