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Position the oven rack to the middle of the oven and preheat to 375 degrees. Brush half of the butter over a jelly roll pan measuring about 17 1/2" x 12 1/2" x 1"h, and line with parchment paper large enough to provide an overhang over the sides of the pan. Cut a line from the corner of the parchment down into the corner of the pan so the parchment fits snugly. Brush the parchment with the other half of the melted butter and set aside.
Place the flour, baking powder, baking soda, pumpkin pie spice, cinnamon, and salt in a medium bowl. Sift the mixture into onto a sheet of waxed paper, then sift once more and set aside.
Place the eggs and sugar in the bowl of a stand mixer fitted with the whisk attachment and whisk to combine. Beat on medium high speed, about 5-8 minutes, or until the mixture has tripled in volume. Dip the whisk into the batter and draw a figure 8 -- if it holds for a few seconds, it's ready. Stir the vanilla into the pumpkin and add it to the egg mixture in about 4 additions, beating on medium high speed until each addition is incorporated.
Sift 1/3 of the flour mixture over the pumpkin mixture and gently fold with a rubber scraper until just incorporated, being careful not to deflate the batter. Sift 1/3 more and fold, repeating until the flour has been incorporated. Pour into the prepared pan and gently spread the batter evenly with a baby offset spatula, being careful not to deflate it. Bake on the middle rack of the oven for 12-15 minutes, or until the cake springs back when gently pressed with your finger. Cool 5 minutes on a wire rack.
Lay a kitchen towel large enough for the cake on the counter. Dust the towel with a generous amount of powdered sugar so that the towel is completely covered. Pick up the cake pan with pot holders, holding it upright from the top with the cake facing away from you, making sure the pot holders are around the top of the cake slightly to keep it in place. Hold the cake at an angle over the sugar dusted kitchen towel, remove your hands and turn the cake out quickly onto the kitchen towel. Remove the parchment paper, and roll the cake in the towel, and place on a wire rack to cool completely.
When the cake has cooled completely, unroll it and trim off the outer edges of the cake with a serrated knife. Set aside.
Place the whipping cream, sugar, and vanilla in the bowl of a stand mixer fitted with the whisk attachment. Beat on medium high just until stiff peaks begin to form. Spread onto the cake in an even layer leaving 1/2" border around the edges. The cream should be spread to about 1/4" thickness, as if there's too much it will ooze out of the sides when rolled -- you will have some cream left over. Roll the cake from the short end, tucking it in as you go. Place on a serving dish and refrigerate for a minimum of 2 hours. Slice, serve, and enjoy!