Preheat the oven to 350°F. Line cookie sheets with parchment paper and set aside.
In a small bowl, stir together flour, pumpkin spice, and baking soda, and set aside.
In a large bowl with an electric hand mixer or in a stand mixer with the paddle attachment, combine the butter, brown sugar, granulated sugar, pumpkin puree, and vanilla until smooth.
Add the flour mixture to the wet mixture and mix until combined, scraping down the sides.
Add in the oats and mix until evenly distributed, the mixture will be thick.
Use a cookie scoop or roll dough into balls and place them 2 inches apart on your prepared cookie sheets.
Bake for 11-13 minutes until they puff slightly, appear dry on top, and lightly golden around the edges.
Allow cookies to cool on the tray for about 2 minutes before transferring to a wire rack to cool completely.
While the cookies are cooling, make the frosting. Whip the butter in a medium bowl with an electric hand mixer until smooth.
Add one spoonful at a time of the powdered sugar until it is all combined, it will resemble sand.
Add 3 tablespoons of the cream, vanilla, and pumpkin spice. Mix until combined, if it still seems too thick, add an additional tablespoon of cream.
Sprinkle pumpkin spice over iced cookies.