Preheat oven to 375F. Line a 10x15 inch jelly roll pan with foil and spray with cooking spray.

Beat eggs at high speed for 5 minutes. Gradually add sugar, applesauce, and molasses.

Stir together flour, baking powder, cinnamon, allspice, nutmeg, and salt. Add to egg mixture and spread into the prepared pan.

Bake for 10 minutes. Turn out onto a kitchen towel sprinkled with powdered sugar. Roll up the towel and cake together. Cool completely.

While the cake is cooling, make the apple filling by melting butter, adding apples, sugar, cinnamon, and nutmeg. Cook for 2 minutes. Mix water and cornstarch, add to pan, cook for 5 minutes, then chill.

Unroll the cake carefully, spread the cooled apple filling evenly, then re-roll the cake. Place on a rack.

Toast chopped nuts if desired, then prepare caramel frosting by melting butter, adding brown sugar, stirring in cream, then powdered sugar and vanilla. Pour over the cake and garnish with nuts.