Cream together the softened butter and brown sugar using a stand mixer or hand mixer until light and fluffy.
Beat in the egg, vanilla extract, and maple syrup. Then, mix in the pumpkin puree until the batter is smooth.
Whisk together the flour, baking soda, pumpkin spice, and salt in a separate bowl. Gradually add the dry ingredients to the wet ingredients, mixing until well combined.
Stir in the oats until they are evenly distributed throughout the dough.
Drop the cookie dough by 2-tablespoon-sized scoops onto a baking sheet. Bake at 350°F for 12-15 minutes, or until the edges are lightly golden. Let the cookies cool on the baking sheet before transferring them to a wire rack to cool completely.
While the cookies cool, whisk together powdered sugar, maple syrup, cinnamon, and milk in a small bowl until the glaze is smooth.
Once the cookies are fully cooled, drizzle the cinnamon maple glaze generously over each one, letting the sweet glaze set and harden for a perfect finishing touch.