Prepare the pan. Preheat oven to 375°F (190°C). Line a 10×15-inch jelly roll pan with parchment paper, then lightly coat with cooking spray.
Beat the eggs and sugar. In a large mixing bowl, use a hand mixer or stand mixer to beat eggs and sugar until thick, pale, and fluffy (about 2–3 minutes).
Add the pumpkin. Mix in pumpkin puree until smooth.
Combine dry ingredients. In a small bowl, whisk together flour, baking powder, baking soda, pumpkin pie spice, and salt.
Make the batter. Gently fold the dry ingredients into the wet mixture until just combined—do not overmix.
Bake. Spread the batter evenly into the prepared jelly roll pan. Bake 12–15 minutes, or until the cake springs back when lightly touched.
Roll the warm cake. Immediately invert the hot cake onto a clean kitchen towel dusted generously with powdered sugar. Peel away parchment paper. Starting at the short end, roll the cake up with the towel inside. Place seam-side down on a wire rack and let cool completely.