Beat brown sugar and butter together in a large bowl with an electric mixer until creamy, 1 to 2 minutes. Beat pumpkin, egg, and 2 teaspoons vanilla extract into butter mixture.
Whisk flour, 2 tablespoons cinnamon, ginger, nutmeg, baking powder, cloves, allspice, and salt together in a bowl. Add flour mixture to pumpkin mixture and stir until dough is completely combined.
Fill a cookie press with pumpkin cookie dough. Press cookies 2-inches apart on baking sheets.
Bake cookies in batches in the preheated oven until edges begin to brown, 5 to 8 minutes. Transfer cookies to wire racks to cool completely.
Whisk confectioners' sugar, milk, corn syrup, 4 teaspoons cinnamon, and 2 teaspoons vanilla extract together in a bowl to an icing consistency. Dip cooled cookies in icing and shake gently to remove excess. Place iced cookies on a sheet of waxed paper while icing dries.
