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. You should have about 7 tablespoons (98 grams) of brown butter.
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, cinnamon, pumpkin spice, baking soda, baking powder, cream of tartar, 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 medium (1½-tablespoon) spring-loaded cookie 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 15-20 minutes.
While the dough is chilling, preheat the oven to 350°F (180°C) and make the rolling mixture. Combine the sugar, cinnamon, and pumpkin spice in a small bowl. Line a baking sheet with parchment paper, then set it aside.
Roll each dough ball in the spiced sugar 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 10 minutes, then transfer to a cooling rack to cool completely.
Once cool enough to handle, dip the baked cookies in the remaining spiced sugar mixture.