Preheat oven to 350°F (175°C). Line a baking sheet with parchment paper.
In a mixing bowl, cream together the butter and brown sugar until light and fluffy.
Add the egg, pumpkin puree, and vanilla extract. Mix until well combined.
In a separate bowl, whisk together the flour, oats, baking soda, cinnamon, nutmeg, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Drop spoonfuls of dough onto the prepared baking sheet, spacing them about 2 inches apart.
Bake for 10-12 minutes, or until the edges are lightly browned.
Allow cookies to cool on the baking sheet for 5 minutes before transferring to a cooling rack to cool completely.