Preheat oven to 350 degrees and grease a9x13 pan. In a large bowl beat 1/2 Cup butter, applesauce, eggs and vanilla. Add in flour, baking soda, baking powder, salt, and brown sugar. Mix until well blended and spread evenly in 9x13 pan.
Mix butter and 3/4 Cup sugar until smooth. Add in pumpkin, flour, baking powder, baking soda, salt, egg, and pumpkin pie spice. Mix until well combined. Spread evenly over snickerdoodle layer. In a small bowl, mix 1/4 cup sugar with 2 teaspoons cinnamon. Sprinkle evenly over pumpkin layer.
Bake at 350 degrees for 35-40 minutes, or until a toothpick comes out clean. Cool completely.
Beat butter with melted white chocolate, powdered sugar, and vanilla until smooth. Spread evenly over cooled cake.