Preheat 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.
Scoop slightly softened ice cream in between two cookies. Wrap each sandwich in plastic wrap and freeze about 45 minutes. Roll sides in nuts, mini chocolate chips or English toffee bits. Keep frozen until ready to eat.
(For cookies with icing) 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.