Preheat the oven to 375 degrees F (190 degrees C). Grease a 10x15-inch jelly roll pan and line the bottom with waxed paper.

In a medium bowl, whisk together the flour, cinnamon, baking powder, ginger, nutmeg, and salt.

In a large bowl, beat the sugar and eggs with an electric mixer until well blended. Beat in the pumpkin puree. Add the flour mixture and stir just until no dry streaks remain. Spread the batter evenly into the prepared pan.

Bake until a toothpick inserted into the center comes out clean, about 15 minutes. Remove from the oven and let the cake cool in the pan for 10 minutes.

Turn the cake out onto a clean kitchen towel. Peel off and discard the waxed paper. Starting from a short end, roll the cake and towel together into a log, then let it cool completely, about 30 minutes.

Meanwhile, make the filling: beat the softened cream cheese, confectioners' sugar, butter, 1 tablespoon pumpkin puree, and vanilla extract in a mixing bowl until smooth.

Lay a large sheet of plastic wrap on a work surface. Carefully unroll the cooled cake onto the plastic wrap.

Spread the filling evenly over the cake. Roll the cake back up without the towel, using the plastic wrap to help. Wrap tightly in plastic wrap, place seam-side down on a plate, and refrigerate until fully chilled, about 2 hours.