Preheat oven to 375 degrees F (190 degrees C). Grease a 10x15-inch jelly roll pan and line with wax paper.
Beat eggs and white sugar in a large bowl until well blended; stir in 2/3 cup pumpkin puree.
Combine flour, cinnamon, baking powder, ginger, nutmeg, and salt in another bowl; stir into egg mixture until just blended. Pour mixture into prepared jelly roll pan.
Bake in preheated oven until a toothpick inserted into the center comes out clean, about 15 minutes. Cool in the pans for 10 minutes, then turn cake out onto a clean towel. Remove and discard wax paper. Roll cake up into the towel, starting with the short end. Cool.
Beat cream cheese, confectioners' sugar, butter, 1 tablespoon pumpkin puree, and vanilla extract in another bowl until smooth.
Spread a large sheet of plastic wrap on a work surface. Place and unroll cake over plastic and spread with prepared filling. Re-roll cake and wrap with plastic. Refrigerate, seam-side down, until chilled, about 2 hours.