Make cookies: Preheat the oven to 375 degrees. Whisk together flour, baking powder, baking soda, salt, cinnamon, ginger, and nutmeg in a medium bowl; set aside.
Put butter and brown sugar in the bowl of an electric mixer fitted with the paddle attachment. Mix on medium speed until pale and fluffy, about 3 minutes. Mix in eggs. Reduce speed to low. Add pumpkin, evaporated milk, and vanilla; mix until well blended, about 2 minutes. Add flour mixture; mix until combined.
Transfer 1 1/2 cups batter to a pastry bag fitted with a 1/2-inch plain tip (such as Ateco #806). Pipe 1 1/2-inch rounds onto parchment-lined baking sheets, spacing 1 inch apart. Bake cookies, rotating sheets halfway through, until tops spring back, about 12 minutes. Cool on sheets on wire racks 5 minutes. Transfer cookies to wire racks; let cool completely. (I top cookies with Turbinado or Cinnamon Sugar before baking), and then I ice them.
In small bowl, beat butter and syrup until blended. Gradually beat in powdered sugar until smooth.