Prick the sweet potato several times with a fork and microwave for 8 minutes, until tender. Cut the potato in half and scoop the flesh out. Mash with a fork (you should have about 1 cup of sweet potato).
Either just melt the butter, or if you wish to take your sweet potato doughnut flavor to the next level, brown the butter in a small skillet. Let it cool.
In a small saucepan, heat the milk to 105 degrees F on an instant read thermometer). It’s important not to go over 105, or you risk killing the yeast (this happened to me the first time I made these…oops).
In the bowl of a stand mixer fitted with the whisk, let the yeast and warm milk sit for 5 minutes. Add the sugar, brown sugar, salt, vanilla seeds, nutmeg, and rum, and whisk to combine. Add the mashed sweet potato, butter, egg and egg yolks, and beat until well combined.
Change to the dough hook, and add the flour. Beat at medium for 2 minutes, then bump the speed up to medium high and knead until a soft dough forms and the sweet potato dough pulls away from the sides and toward the center, about 5-7 minutes.
Dump the dough out, form it into a ball, and let it rise on the countertop for 2 hours (don’t be alarmed if it doesn’t rise all that much…it won’t).
Gently redistribute the yeast by kneading lightly, then roll the dough out onto a lightly floured surface. I rolled my dough out to about 3/4 inch thick for my doughnuts.
Cut circles out, then stamp out the centers to make little holes. Place the doughnuts on two parchment paper lined sheet pans, and let them rise for 2 hours, covered with plastic wrap.
Preheat the oven to 400 degrees F. Bake the doughnuts in the center of the oven for about 10 minutes, until the doughnuts have risen and are golden brown (they should be 200 degrees F on an instant read thermometer).
While the doughnuts are baking, make the cinnamon sugar topping by melting the butter in one bowl, and whisking the sugar and cinnamon in another. While the doughnuts are hot out of the oven, toss them in the butter, then immediately toss in the cinnamon sugar.