Preheat the oven to 375°F. Line a 17-inch cookie sheet with parchment paper and lightly spray with non-stick cooking spray.
Cover a large cutting board with parchment paper and set it aside.
In a medium bowl, combine flour, baking powder, baking soda, pumpkin pie spice, and salt.
In a large bowl with an electric mixer, beat eggs, vanilla, and sugar until thick. Add in the pumpkin, and mix to combine.
Stir the flour mixture into the pumpkin mixture.
Spread the batter evenly into the prepared pan. Bake for 13 to 15 minutes, or until the top of the cake springs back when touched.
When you take the cake out of the oven, immediately turn the cake onto the parchment paper-covered cutting board. It's easiest to put the parchment paper and cutting board on top of the cake and flip them together in one swift movement. Then gently loosen and peel off the parchment paper that was baked with the cake.
Immediately, while the cake is still hot, roll up the cake and the parchment paper together, starting at the narrow end. Cool completely on a wire cooling rack (about 2 hours).