Preheat oven to 350 degrees F. Prepare a cookie sheet with non stick spray or parchment paper and set aside.
Start my whisking together your dry ingredients, flour, oats, salt, baking soda, and pumpkin pie spice. Set aside.
In a stand mixer or in a large bowl with a hand mixer cream together butter and sugars (white and brown). Mix until creamy, smooth, and well combined. Add egg, vanilla, and pumpkin puree. Slowly add in dry ingredients mixing until combined.
Dough is very sticky so it will need to chili. Place in the fridge to chili for 1-2 hours or until dough can be formed into balls.
Roll out cookie dough balls and place on prepared baking sheet. Bake for 12-13 minutes. Allow cookie to cool on baking sheet for 5-8 minutes before removing to cool on a wire rack.
While cookies bake prepare maple glaze. Whisk all ingredients together until smooth and well combined. Add to the top of cooks and allow to set for 3-5 minutes before serving.