1. Make the doughnuts: Prick the sweet potato all over with a fork and cook it in a microwave at high power for 10 minutes, until tender. Let cool, then peel and puree the sweet potato; you should have about 1 cup.
2. In a small skillet, cook the butter over moderate heat until nutty and lightly browned, about 4 minutes. Scrape the browned butter and solids into a small bowl and let cool.
3. In the same skillet, heat the milk until just warm, about 105 degrees. Pour the warm milk into the bowl of a standing electric mixer fitted with the dough hook. Add the yeast and let stand for 5 minutes. Gently mix in the granulated sugar, light brown sugar, salt, vanilla seeds, nutmeg and rum. Add the sweet potato puree, browned butter and solids, egg and egg yolks and beat until combined. Add the 3 1/4 cups of bread flour and beat at medium speed until the dough is evenly moistened, about 2 minutes. Increase the speed to moderately high and beat until a soft dough forms, about 5 minutes. Gather the dough into a ball and transfer to a buttered bowl. Cover and let rise in a draft-free place for 1 hour.
4. Punch down the dough and let stand for 5 minutes. On a lightly floured surface, roll out the dough 1/2 inch thick. Using a 2 3/4-inch round cutter, stamp out as many rounds as possible. Using a smaller round cutter (1 inch), stamp out the centers. Transfer the doughnuts and holes to 2 parchment paper–lined baking sheets. Cover loosely with plastic wrap and let the doughnuts and holes rise in a warm place for 1 hour.
5. Preheat the oven to 400 degrees and position racks in the upper and lower thirds. Bake the holes for 10 minutes and the doughnuts for about 20 minutes, until risen and golden.
6. Meanwhile, make the topping: In a small bowl, combine the sugar and cinnamon. Put half of the hot doughnuts in a large bowl and drizzle with some of the melted butter; toss and turn to coat. Sprinkle with some of the cinnamon sugar and toss and turn until evenly coated. Repeat with the remaining doughnuts, butter and cinnamon sugar. Transfer the doughnuts to a platter; serve.