These soft, lightly spiced cake-like cookies are studded with minced candied ginger and topped with a buttery sugar glaze.
Preheat the oven to 350°. 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.
Scoop level tablespoons of the batter onto a greased baking sheet, 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.