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.
Make icing: Put confectioners' sugar in a large bowl; set aside. Melt butter in a small saucepan over medium heat. Cook, swirling pan occasionally, until golden brown, about 3 minutes. Immediately add butter to confectioners' sugar, scraping any browned bits from sides and bottom of pan. Add evaporated milk and vanilla; stir until smooth. Spread about 1 teaspoon icing onto each cookie. If icing stiffens, stir in more evaporated milk, a little at a time.
Cookies can be stored in single layers in airtight containers at room temperature up to 3 days.