In a medium bowl, whisk together flour, baking soda, baking powder, salt, cinnamon, ginger, nutmeg, and allspice; set aside.
In a large bowl, whisk together, brown sugar, granulated sugar, butter, and eggs. Add dry ingredients, and whisk until smooth. Whisk in sweet potato purée.
Divide batter evenly among liners, filling each about halfway. Bake until tops spring back when touched, and a cake tester inserted in the center comes out clean, 20 to 25 minutes, rotating pans once if needed. Transfer to a wire rack; let cool.
Frosting Instructions
In a saucepan over medium heat, melt the butter.
Add milk and brown sugar, stir to combine.
Boil vigorously for 1 minute.
Remove from heat and beat in 1/2 cup powdered sugar.
Cool slightly then and beat in the vanilla and remaining powdered sugar, add more half in half if necessary.
Quickly spoon over cupcakes before glaze sets.
Notes
Adapted from Martha Stewart's Pumpkin Cupcakes.See post for filling and topping instructions