Brown the butter in a light-colored saucepan over medium heat. Swirl occasionally until butter foams, then develops golden brown specks and a nutty aroma (about 5-7 minutes). Immediately transfer to a large bowl and let cool for 10 minutes.
Preheat your oven to 350°F (175°C) and line baking sheets with parchment paper.
Whisk both sugars into the cooled brown butter until well combined and no lumps remain. The mixture should look like wet sand.
Add the pumpkin puree, egg, and vanilla to the butter-sugar mixture and whisk until smooth and well incorporated.
In a separate bowl, combine the flour, oats, baking soda, salt, and pumpkin pie spice, whisking to distribute the spices evenly.
Gradually add the dry ingredients to the wet mixture, stirring just until no flour streaks remain. Over-mixing will make tough cookies!
Fold in the chocolate chips (and nuts if using) until evenly distributed throughout the dough.
Chill the dough for at least 30 minutes (or up to 24 hours) to enhance flavor and prevent excessive spreading.
Scoop dough using a 2-tablespoon cookie scoop and place balls 2 inches apart on prepared baking sheets.
Bake for 11-13 minutes until edges are set and golden but centers still look slightly soft. They’ll continue cooking on the hot baking sheet.
Let cookies cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.