Preheat oven to 400F. Start on the filling by washing and drying the sweet potato, then piercing with fork all over. Lightly rub a little oil over the entire surface of the sweet potato, then place on a parchment-lined sheet pan. Bake for about 40 minutes, or until a toothpick or skewer is inserted into the center of the potato with no resistance. Remove from the oven and set aside to cool slightly.
Meanwhile, for the cookie dough, in a small 1-quart saucepan, add unsalted butter and cook over medium low until it melts and starts to bubble. Continue cooking until bubbling subsides and butter turns brown and smells fragrant and nutty, about 6-8 minutes. Remove from the heat, set aside and allow to cool for about 10 minutes.
In a small bowl, whisk together flour, cream of tartar, baking soda, salt, and cinnamon until combined. Set aside.
When browned butter is cool, combine cooled butter and sugar in stand mixer fitted with a beater attachment. Mix on medium speed until combined and smooth, 3 minutes.
Add in egg and allow to incorporate completely. Scrape down bottom and sides of bowl and mix again for 30 seconds.
Reduce the speed of the mixer to low, then add in your flour mixture and mix until just combined. Scrape down the bottom and sides of bowl. Add in vanilla and mix for a final 30 seconds.
Once combined, cover the bowl with plastic wrap and set in refrigerator until dough has firmed up, but is still easily shaped, about 1 hour.
While the cookie dough chills, return to the sweet potato filling. While the sweet potato is still warm, carefully remove the potato from its skin and discard. Place the potato in a large bowl and whisk in salted butter and sugar for 1 minute. (Once you start to mix the baked potato, long little stringy pieces will attach to the whisk; you'll want to rinse the whisk to get rid of those. Removing them makes the filling super smooth, so you're not eating the stringy parts.) Repeat these two or three times, each time rinsing the strings from the whisk.
Whisk in the egg and vanilla extract, followed by the nutmeg, cinnamon, and flour until your filling mixture is smooth. Cover with plastic wrap, until the cookie dough has finished chilling.
Reduce oven to 350F. Prepare two sheet pans each with parchment paper.
For the rolling mixture, combine all ingredients in a small bowl and stir together.
To assemble cookies, remove cookie dough and filling from the refrigerator. Using a #40 scoop, scoop out cookie dough, press it into a small 2-inch disk, then roll it in the rolling mixture. Place 6 cookies on each sheet pan, spread evenly apart.
Then use the scoop to press in the center of each cookie, leaving an indentation for the filling. Place about a tablespoon of filling in the middle of each cookie.
Bake until the filling has softened, and the cookie has spread and just started to brown at the bottom and edges of the cookie, about 13-15 minutes. Remove from the oven and let cool for 2 minutes on the pan, then move to a wire rack to continue cooling for another 3-5 minutes. Dust lightly with powdered sugar and enjoy warm!