Preheat your oven to 350°F (175°C).
Brown the butter in a saucepan over medium heat until it turns golden brown.
In a large bowl, combine the brown butter, brown sugar, and granulated sugar.
Add the pumpkin puree, egg, and vanilla extract to the sugar mixture, and beat until smooth.
In a separate bowl, whisk together the flour, baking soda, cinnamon, nutmeg, and salt.
Gradually add the dry mixture to the wet mixture, stirring until just combined.
Fold in the chopped walnuts, if using.
Scoop tablespoon-sized balls of dough onto a parchment-lined baking sheet, spacing them about 2 inches apart.
Bake for 10-12 minutes or until the edges are lightly golden.
Let the cookies cool on the baking sheet for 5 minutes before transferring to a wire rack.
