Preheat oven to 350 degrees F (175 degrees C). Line 12 muffin cups with cupcake liners.
Beat butter and brown sugar with an electric mixer in a large bowl until light and fluffy. Add room-temperature eggs one at a time, allowing each egg to blend into the butter mixture before adding the next. Blend in vanilla extract and sweet potatoes.
Whisk flour, baking powder, baking soda, 1/2 teaspoon salt, cinnamon, ginger, nutmeg, and cloves in a bowl. Add half the flour to the sweet potato mixture, stirring just until incorporated. Blend in the milk and the remaining flour mixture.
Scoop batter into prepared cupcake pan and bake in preheated oven until tops spring back when touched lightly with a finger and a toothpick inserted in the center of a cupcake comes out clean, 18 to 22 minutes. Cool on rack.
To make frosting, combine white sugar, cream of tartar, a pinch of salt, egg whites, and cold water in a heatproof mixing bowl. Set the mixing bowl over a pan of simmering water and beat with an electric mixer until the mixture is very hot to the touch and stiff peaks have formed, 5 to 7 minutes. Remove the bowl from the heat and beat 1 minute more. Add 1 teaspoon vanilla extract and marshmallow creme and beat until combined.
To frost the cupcakes, fill a pastry bag fitted with a large plain tip with frosting; pipe mini marshmallow-shaped dots all over the cupcakes. Alternatively, use a knife and frost them generously with lots of swoops and swirls.
Set oven rack about 6 inches from the heat source and preheat the oven's broiler.
Arrange 3 or 4 cupcakes on a baking sheet and place them in the oven under the broiler. Toast until the frosting has started to brown, about 90 seconds (check every 20 seconds and rearrange the baking sheet, if necessary). Repeat with the remaining cupcakes until all are toasted.
