Special equipment: Three 12-cup mini muffin tins or two 24-cup mini muffin tins
Make the sweet potato puree: Preheat the oven to 350 degrees F. Roast the sweet potatoes until very tender, about 1 hour. Let cool, peel, then puree in a food processor until smooth. Set aside 1 cup of the puree for the cupcakes and reserve the remaining puree for another use.
Make the cupcakes: Position an oven rack in the center of the oven, and preheat to 350 degrees F. Line mini muffin tins with paper liners, and coat the liners and the top of the tin with cooking spray. Halve each marshmallow on the diagonal with kitchen shears; set aside.
Whisk the flour, cinnamon, ginger, baking powder, baking soda, salt and cayenne together in a large bowl. Beat the butter, brown sugar and granulated sugar in another large bowl with an electric mixer on medium-high speed until light and creamy. Add the 1 cup of reserved sweet potato puree, and beat until incorporated (it's OK if the batter curdles). Beat the eggs in 1 at a time. Beat in the vanilla. Adjust the mixer speed to medium-low, and beat in the flour mixture in 3 additions, scraping down the side of the bowl as needed, until just incorporated.
Divide the batter evenly among the cups of the prepared muffin tins. Bake until the tops spring back when pressed, 18 to 20 minutes, rotating the tins halfway through. Let the muffins cool in the tins for a few minutes, then transfer them to a rack. While the cupcakes are still warm, top each with a marshmallow half, cut-side down. Let the cupcakes cool completely. (They can be cooked and cooled a day ahead and stored at room temperature in an airtight container.)
Preheat the broiler. Transfer half the cupcakes to an unlined baking sheet, and broil, rotating the sheet as needed, until the marshmallows are toasted and golden. Repeat with the remaining cupcakes. This can be done up to 2 hours before serving.
If you own a kitchen torch, feel free to use that to toast the marshmallows.
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