Preheat the oven to 375 degrees. Line baking sheets with parchment paper; set aside. Fit a pastry bag with a large coupler and a 1/2-inch plain round (such as Ateco #806); set aside.
Whisk together flour, baking powder, baking soda, salt, cinnamon, ginger, and nutmeg in a medium bowl; set aside. Cream 12 tablespoons (1 1/2 sticks) butter and the brown sugar on medium speed in the bowl of an electric mixer fitted with the paddle attachment until pale and fluffy, about 3 minutes.
Mix in eggs. Reduce speed to low.
Add pumpkin, 3/4 evaporated milk, and 1 teaspoon vanilla; mix until well blended, about 2 minutes.
Add flour mixture; mix until combined.
Transfer 1 1/2 cups batter to a pastry bag. 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.
Put confectioners' sugar in a large bowl; set aside. Melt remaining 10 tablespoons 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 remaining 5 tablespoons evaporated milk and remaining 2 teaspoons 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.