Preheat oven to 375F. Line a jelly roll (10x15") pan with foil and spray with cooking spray. I like to use the spray with flour in it.
Beat eggs at high speed for 5 minutes. Gradually beat in sugar, applesauce, and molasses.
Stir together flour, baking powder, cinnamon, allspice, nutmeg, cloves and salt. Add to other mixture. Spread into prepared pan.
Bake for 10 minutes. Turn out immediately onto a kitchen towel sprinkled liberally with powdered sugar. Starting at the narrow end, roll towel and cake together. Cool completely.
While the cake is baking and cooling, make the filling. Melt butter in a medium frying pan. Add apples, sugar, cinnamon, and nutmeg. Stir to combine. Cook for about 2 minutes, until apples start to release their juices. Stir together water and cornstarch and add to pan. Stir, and continue cooking over medium-low heat for about 5 minutes, until liquids are absorbed and apples are cooked. Remove from heat, then place apples in a bowl and chill until ready to assemble cake.
Once cake and apples are cool, carefully unroll the cake. Spread apples evenly across the cake, being sure to spread to the edges as much as possible. Starting at the narrow end, re-roll the cake. Place on a rack set over a cookie sheet to await frosting.
Toast chopped nuts in a small frying pan over low heat, stirring often, for just a few minutes. DO NOT WALK AWAY from the stove. They can burn in an instant. You can also toast them in a 350 degree oven on a cookie sheet, shaking the pan occasionally, for just a few minutes. Same thing: don’t walk away! Cool while preparing frosting.
To make the frosting, melt the butter in a medium sauce pan. I use salted butter, if you use unsalted, add a pinch of salt.
Stir in brown sugar and cook over medium-low heat, stirring constantly, for 2 minutes. Stir in heavy cream and stir until the mixture comes to a boil (it will probably be boiling almost instantly). Remove from heat and whisk in powdered sugar until no lumps remain, then whisk in vanilla extract. If frosting is too thick to pour at this point, add more cream (1 tablespoon at a time). Mine was pourable as written.
Pour frosting over cake roll evenly, being sure to get the ends and sides. Working quickly, use an offset spatula to help spread the frosting where it’s missing. The excess frosting will drip onto the cookie sheet. You can discard that, or eat it with a spoon. Sprinkle nuts over the top, if desired, to make it look prettier for serving. Carefully transfer to serving platter. Store in refrigerator, but serve at room temperature.