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

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

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

Bake for 10 minutes. Turn out immediately onto a sugared towel and roll while warm to cool.

Cook apple filling: melt butter, add apples, sugar, spices, cook 2 minutes, add water and cornstarch, cook 5 minutes, chill.

Unroll cooled cake, spread apple filling, re-roll, and cool on rack.

Toast nuts in a pan or oven and set aside.

Prepare frosting: melt butter, stir in brown sugar, add cream, boil, whisk in powdered sugar and vanilla.

Pour frosting evenly over the cake roll, sprinkle nuts, and serve at room temperature.