Preheat your oven to 350°F. Line two baking sheets with parchment paper.
In a medium bowl, combine the rolled oats, flour, baking soda, cinnamon, nutmeg, and salt. Set aside.
In a large bowl, cream together the softened butter, brown sugar, and granulated sugar until light and fluffy. Add the egg and vanilla extract, mixing until well combined.
Stir in the pumpkin puree until fully incorporated. Gradually add the dry ingredient mixture to the wet ingredients, mixing until just combined.
Using a cookie scoop or tablespoon, drop rounded balls of dough onto the prepared baking sheets, spacing them about 2 inches apart.
Bake for 12-15 minutes, or until the edges are lightly golden. Remove from the oven and let cool on the baking sheets for 5 minutes before transferring to a wire rack to cool completely.
While the cookies cool, prepare the cream filling. In a medium bowl, beat together the softened butter and powdered sugar until smooth. Add the vanilla extract and enough milk to achieve a spreadable consistency.
Once the cookies are completely cool, spread a generous amount of cream filling on the flat side of one cookie and top with another cookie, pressing gently to sandwich them together.
Serve immediately or store in an airtight container in the refrigerator for up to a week.