Preheat the oven to 350°F (175°C). Grease two baking sheets.
In a bowl, whisk together flour, baking powder, cinnamon, baking soda, nutmeg, and ginger.
Using an electric mixer, beat butter, white sugar, brown sugar, vanilla, egg, and pumpkin puree in a separate large bowl until smooth. Gradually stir the dry ingredients into the pumpkin mixture until combined; batter will be moist.
Drop spoonfuls of batter about 2 inches apart onto prepared baking sheets.
Bake for 10 to 12 minutes or until lightly browned, switching racks halfway through.
Remove from oven and let cool on sheets for 5 minutes. Then transfer cookies to a wire rack and cool completely (about 25 minutes).
For the frosting: Using an electric mixer, beat cream cheese, butter, and vanilla in a bowl until soft and creamy. Gradually beat in powdered sugar (about 1/2 cup at a time) until frosting is smooth and spreadable.
Spread frosting over cooled cookies.
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