Preheat your oven to 350°F (175°C). Line two large baking sheets with parchment paper or silicone baking mats to prevent sticking and ensure even browning.
In a large mixing bowl, beat together the softened butter, white sugar, and brown sugar until the mixture is light and fluffy. This should take about 2 to 3 minutes using a hand mixer or stand mixer.
Add the pumpkin puree, egg, and vanilla extract to the butter mixture. Beat well until everything is fully combined and smooth. Don’t worry if the mixture looks slightly curdled; it will come together once the dry ingredients are added.
In a separate medium bowl, whisk together the flour, rolled oats, cinnamon, pumpkin pie spice, baking soda, and salt. Mixing them beforehand ensures the spices and leavening agents are evenly distributed.
Gradually add the dry ingredients to the wet ingredients. Mix on low speed or by hand with a spatula until just combined. If you are using chocolate chips or raisins, fold them in gently at this stage.
Drop rounded tablespoons of dough onto the prepared baking sheets, spacing them about 2 inches apart. Bake for 10 to 12 minutes, or until the edges are lightly golden and the centers are set. Let the cookies cool on the baking sheet for 5 minutes before transferring them to a wire rack.