Preheat the oven to 350°F (175°C) and line a baking sheet with parchment paper.
Beat the butter, granulated sugar, and brown sugar until light and fluffy.
Add the pumpkin purée, egg, and vanilla. Mix until well combined.
In a separate bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, nutmeg, and ginger.
Gradually add the dry ingredients to the wet mixture, mixing just until combined.
Drop rounded spoonfuls of dough onto the prepared baking sheet, spacing them about 2 inches apart.
Bake for 12–15 minutes, until the edges are lightly golden. Transfer to a wire rack and let cool completely.
For the frosting, beat the butter until smooth and creamy. Add the softened cream cheese and beat until fully combined.
Gradually mix in the icing sugar, vanilla, and a pinch of salt until light and fluffy.
Spread the frosting over the completely cooled cookies before serving.