Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper.
In a large mixing bowl, cream together the softened butter, brown sugar, and granulated sugar until light and fluffy.
Beat in the egg, vanilla extract, and pumpkin puree until smooth.
In a separate bowl, whisk together the flour, oats, pumpkin pie spice, cornstarch, baking soda, baking powder, and salt.
Gradually mix the dry ingredients into the wet mixture until just combined. Do not overmix.
Scoop 2-tablespoon portions of dough onto prepared baking sheets, spacing about 2 inches apart.
Bake for 11–13 minutes, or until the edges are set and the centers are soft but not wet.
Let cookies cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.
For the icing, whisk together powdered sugar, milk, and vanilla until smooth. Drizzle over cooled cookies.
Allow the icing to set before serving.