Heat oven to 375 degrees F.
Line a 10"x15" pan with parchment paper.
Spray lightly with cooking spray. Set aside.
In a small bowl, combine flour, cornstarch, and pumpkin pie spice. Set aside.
In the bowl of an electric mixer fitted with the whisk attachment, beat eggs and sugar until light and fluffy, about 10 minutes.
Mix in pumpkin.
Sift flour mixture over pumpkin mixture.
Stir to combine.
Remove 1 cup of batter and combine it with vegetable oil, mixing well.
Stir back into sponge.
Pour into prepared pan, making sure to get batter all the way to edges and in corners.
Tap to level and remove air bubbles.
Bake for 12 minutes, or until cake is just done.
While cake is baking, lay out a clean dish towel and top it with a piece of parchment paper.
When cake comes out of the oven, sift powdered sugar over top.
Immediately invert it onto parchment paper.
Carefully remove pan and peel parchment off of baked cake.
Beginning with short end, roll cake and parchment up together jelly roll-style.
Set aside to cool for 15 minutes.
Allow to cool before filling.