Using a stand or hand mixer, cream together the brown butter, brown sugar, and granulated sugar until smooth and creamy.
Add in the egg and vanilla bean paste and beat until the texture is light and fluffy (about 3-5 minutes, it should look much paler than before). Then beat in the pumpkin puree.
Add in the flour, pumpkin spice, cream of tartar, baking soda, and salt and mix until thoroughly combined. The dough should be wet and sticky.
In a small bowl, whisk together the granulated sugar and pumpkin spice for the topping.
Scoop a generous 3 tablespoons of dough and roll into balls. I use a size 20 cookie scoop. Then gently roll the cookie dough balls in the pumpkin spice sugar mixture.
Cover the dough balls and chill in the fridge overnight for best results. If you absolutely can't wait, just chill it for at least 2 hours.
Preheat your oven to 350°F and line a light colored baking sheet with parchment paper.
Place dough balls at least 2-3 inches apart on the baking sheet (they will spread, I usually only do 6 cookies on a half sheet pan). Place the rest of the cookie dough balls back in the fridge while the first batch cooks.
Bake the pumpkin snickerdoodles for on the center rack of the oven for 9-11 minute or until the edges are set and the centers are still soft.
Remove from the oven and let rest on the baking sheet for 2-3 minutes, then transfer to a cooling rack.
Notes
To make brown butter, melt unsalted butter over medium heat, stirring frequently. Watch closely for the butter to develop brown specs and a nutty aroma. Once you start seeing brown specs, let cook for just a few seconds longer (watch very carefully or it will burn). Remove from the heat and pour into a small bowl so that it doesn't continue to cook. Set aside and let cool for about 20 minutes in the fridge until soft and pliable. I recommend making the brown butter the night before you make the cookies, so that it has time to fully cool to room temperature! Note: For every ½ cup (1 stick or 113g), you'll lose about 1 tablespoon (14g) to evaporation while cooking. So you can either add in an extra 1 tablespoon before browning, or if you're browning big batches, just make up the weight difference in your recipe with regular unsalted butter if needed.Store cookies in an airtight container at room temperature for up to 4 days.Freeze pre-rolled cookie dough balls in the freezer for up to 6 months. I recommend freezing before rolling in pumpkin spice sugar.