Preheat oven to 350 degrees F (175 degrees C). Lightly grease baking sheets.
Whisk flour, baking powder, cinnamon, baking soda, nutmeg, and ginger together in a bowl. Beat 1 cup butter, white sugar, brown sugar, 2 teaspoons vanilla extract, and egg with an electric mixer in a separate large bowl, beating until mixture is smooth. Beat in pumpkin puree. Gradually stir dry ingredients into pumpkin mixture. Batter will be moist.
Spoon batter by teaspoonfuls about 2 inches apart onto prepared baking sheets.
Bake in the preheated oven until cookies are lightly browned, 10 to 12 minutes. Let cookies cool for about 5 minutes on sheets before removing to finish cooling on waxed paper.
Beat cream cheese, 1/4 cup butter, and 1 teaspoon vanilla extract in a bowl with an electric mixer until soft and creamy. Beat in confectioners' sugar, about 1/2 cup at a time, until frosting is smooth and spreadable. Frost cooled cookies with cream cheese frosting.