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. 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. Note: you want to make sure you cake is done. A toothpick should come out completely clean. Err on the side of overdone for cake rolls, that makes them roll easier. Turn out immediately onto a kitchen towel sprinkled liberally with powdered sugar. Starting at the narrow end, roll towel and cake together. Cool completely. Note: the cake is sticky, soft, and spongey. It might come off a little on your fingers, but that’s okay. Cooling will take at least 2 hours. You can make the cake a day ahead. Just wrap the cooled cake with plastic wrap overnight.
Make the filling: Beat 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. (This takes about 2 minutes with my stand mixer on medium speed.) Add powdered sugar, and vanilla. Add eggnog 1 tablespoon at a time, slowly. Continue beating until stiff peaks form. Be careful not to overwhip.
Once cake is cool, carefully unroll it. It might leave a little mess on the towel, but that’s okay, as long as you can peel it off. 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. As you roll it the filling will shift a little and you may need to remove some if you’ve put too much on the cake. That’s okay, just have a knife handy to scrape off any excess as you roll. At this point you can wrap the cake in plastic wrap and refrigerate for up to 24 hours. When ready, place cake on serving plate and top with some of the remaining whipped cream and dust with cinnamon.