Preheat the oven to 350° and line 3 baking sheets with parchment paper. In a small bowl, whisk the flour with the baking powder, baking soda, pumpkin pie spice and salt. In a large bowl, using a handheld electric mixer, beat the butter with the sugar at medium speed until light and fluffy, about 2 minutes. Beat in the egg and vanilla followed by the pumpkin puree and candied ginger. At low speed, beat in the dry ingredients until evenly incorporated.
Using a small ice cream scoop, scoop level tablespoons of the batter onto the prepared baking sheets, 1 inch apart. Using a lightly moistened finger, smooth the tops. Bake the cookies, 1 sheet at a time in the center of the oven, until risen and firm, about 15 minutes.
In a small bowl, combine the confectioners’ sugar with the butter and milk and stir until smooth and spreadable. While the cookies are still slightly warm, spoon a teaspoon of the glaze onto each one and spread slightly. Let cool completely, then transfer to a platter.