Preheat the oven to 350°F (175°C) and line baking sheets with parchment paper.
In a mixing bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, nutmeg, and ginger. Set aside.
In another large bowl, beat the softened butter and granulated sugar together using an electric mixer until light and fluffy, about 2-3 minutes.
Add the egg, pumpkin puree, and 1 teaspoon vanilla extract to the butter mixture. Mix until well combined.
Gradually add the dry ingredients to the wet ingredients, mixing just until combined. Do not overmix.
Using a spoon or your hands, scoop out the dough and form it into balls (about 1 inch in diameter) and place them on the prepared baking sheets, spacing them about 2 inches apart.
Bake in the preheated oven for 10-12 minutes, or until the edges are lightly golden. Remove from the oven and let them cool on the baking sheets for about 5 minutes before transferring them to a cooling rack.
While the cookies are cooling, prepare the frosting by whisking together the powdered sugar, milk, and 1 teaspoon vanilla extract until smooth.
Once the cookies have completely cooled, spread a generous layer of frosting on top of each cookie.