Preheat the oven to 375 degrees F and line a 10 x 15 inch jelly roll pan with parchment paper, with about a 1 inch overhang for easy removal after baking.
In a large bowl, whisk together the pumpkin puree, applesauce, sugar and vanilla until well combined.
To the bowl with the pumpkin mixture, add ingredients in this order: flour, baking powder, baking soda, pumpkin pie spice and salt, then stir with a spoon or spatula until just combined. Be careful not to over mix.
Pour the batter into the prepared pan. Bake for 14-15 minutes, until a toothpick inserted in the middle comes out clean.
Remove from the oven. On a large surface, lay out a large clean dish towel, one with no texture if possible. Sprinkle liberally with powdered sugar. With one swift motion, flip the cake from the pan to the towel. Be confident and you will be fine! Alternatively, you can simply roll it up in the parchment paper, but mine cracked a bit when I tried it that way.
Peel off the parchment paper that is still stuck to the cake. Roll the cake up from one short end to the other end, carefully and using the towel. (SEE PHOTOS ABOVE FOR REFERENCE)
Let the cake cool completely for about 3 hours on a wire rack. Or stick it in the refrigerator to cool a bit faster.