Combine the flour, baking powder, baking soda, cinnamon, nutmeg, ginger, and salt in a bowl and whisk to blend thoroughly.
In a mixing bowl with an electric mixer, beat the grown and granulated sugars with 1/2 cup of softened butter until light and fluffy. Add the pumpkin purée, egg, and 1 teaspoon of vanilla extract. Beat until smooth.
Add the dry ingredients and beat on low speed to combine. Increase the mixer speed and beat until well blended.
Cover the bowl and chill the cookie dough for 1 hour.
Heat the oven to 350 F. Line cookie sheets with parchment paper or baking mats.
Drop the cookie dough onto the prepared baking sheets by rounded tablespoon.
Bake the cookies for 15 to 18 minutes or until the edges begin to brown. Cool slightly in the pan on a rack, then remove the cookies and place them on a rack to cool completely.
For the icing, combine the softened butter with the confectioners’ sugar, caramel topping, 1/2 teaspoon vanilla, and 1 tablespoon of milk in a bowl. Beat until smooth, adding more milk as needed for drizzling or spreading consistency.