Preheat the oven to 350°F and line baking sheets with parchment paper.
In a large bowl,use an electric mixer on medium low speed to cream together the softened butter and brown sugar until light and fluffy.
Add the pumpkin puree, egg, and vanilla extract and mix until combined.
In a separate bowl, combine the oats, flour, baking soda, and pumpkin pie spice.
Gradually add the dry mixture to the wet mixture and stir until just combined.
Drop 2 Tablespoon-sized portions of dough onto prepared baking sheets, spacing them about 2 inches apart.
Bake for 12–14 minutes, or until the edges are lightly golden, then transfer to a wire rack to cool completely.
In a small bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth.
Dip the tops of the cooled cookies into the icing or drizzle over cookies using a piping bag or spoon, then allow the icing to set before serving.