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.
Pumpkin layer
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.
Frosting
Beat butter with melted white chocolate, powdered sugar, and vanilla until smooth. Spread evenly over cooled cake.