Preheat oven to 350 degrees Fahrenheit (175 degrees Celsius) and line baking sheets with parchment paper.
Cream together softened butter, granulated sugar, and brown sugar in a large mixing bowl until light and fluffy.
Incorporate the pumpkin puree, egg, and vanilla extract, beating until smooth.
In a separate medium bowl, whisk together the flour, oats, cinnamon, ginger, cloves, nutmeg, baking soda, and salt.
Gradually fold the dry ingredients into the wet mixture until no flour streaks remain.
Portion dough into 2-tablespoon rounds and place on the prepared baking sheets 2 inches apart.
Bake for 10 to 12 minutes until edges are set but the center remains soft.
Cool cookies on the baking sheet for 5 minutes then transfer to a wire rack to cool completely.
Whisk together powdered sugar, milk, and additional cinnamon to create a glaze.
Drizzle the glaze over the cooled cookies and allow to set before storage.