Line a 10x15 inch jelly roll pan with foil (for easy removal).
Grease and flour the pan (I use the cooking spray that has flour in it for this step.)
You can also line your pan with parchment paper instead of foil; Grease it just in case.
Beat eggs at high speed for 5 minutes, then gradually beat in sugar, applesauce, and molasses.
Stir together flour, baking powder, cinnamon, allspice, ginger, and salt; add to other mixture. Spread into pan.
Bake at 350°F for 10-12 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; Cooling will take at least 2 hours or place in fridge to speed the process.
Make the filling by beating cold heavy whipping cream in a stand mixer with the whisk attachment (or in a large bowl with a hand mixer) until soft peaks form.
Add powdered sugar, and vanilla.
Add eggnog 1 tablespoon at a time, slowly.
Continue beating until stiff peaks form.
Once cake is cool, carefully unroll it.
Spread about half the whipped cream over the top of the cake, leaving an edge at the end.
Starting at the narrow end, re-roll the cake.
At this point you can wrap the cake in plastic wrap and refrigerate for up to 24 hours.
When ready to serve, place cake on serving plate and top with some of the remaining whipped cream and dust with cinnamon.