Cube the butter and place it in a light-colored skillet. Melt the butter over medium heat, stirring constantly. Once melted, the butter will start to foam. Keep stirring. After 5-8 minutes, the butter will begin browning; you'll notice lightly browned specks begin to form at the bottom of the pan, and you'll start to smell a nutty aroma. Once browned, immediately remove from the heat, and pour into a heatproof bowl.
Add the granulated sugar, brown sugar, pumpkin puree, and vanilla to the brown butter. Using a whisk, whisk together until combined, about 1 minute.
Add the flour, pumpkin spice, baking soda, baking powder, and salt into the wet ingredients and stir using a silicon spatula until everything is combined. The dough will be thick and a little greasy.
Using a 1½-tablespoon spring-loaded scoop, scoop the dough into balls and place dough balls on a lined baking sheet, loosely cover with plastic wrap, and place in the refrigerator to chill for 20-30 minutes.
Preheat the oven to 350°F (180°C).
Roll each dough ball in sugar, if using, and bake for 10-12 minutes or until lightly browned on the sides. Remove from the oven and allow to cool on the baking sheet for 5 minutes, then transfer to a cooling rack to cool completely.