Poke the sweet potato all over with a fork. Heat in the microwave in 5-minute increments until cooked through, about 10 minutes. Once cool enough to handle, remove flesh and mash with a fork.
Preheat oven to 350 degrees and line two muffin pans with cupcake liners. Set aside. In a large bowl, using an electric mixer, whisk eggs then slowly add melted butter and milk. Add one cup of sweet potato puree then blend to completely incorporate.
Combine organic cake mix and brown sugar in a large bowl. Add the dry ingredients to the wet mixture and combine with the mixer on medium-low speed. Once incorporated, increase speed to medium-high and beat one minute.
Fill cupcake liners 2/3 full, then bake 16-18 minutes, or until a toothpick inserted in the center comes out clean. Remove from oven and cool completely on a wire rack.
To make the frosting: Bring 1” of water to a simmer in a small saucepan. Whisk together egg whites, cane sugar and white vinegar in a medium sized glass bowl. Set bowl over saucepan, making sure the bowl does not touch the water.
Use an electric hand blender on medium speed to beat constantly, until the sugar dissolves and the egg mixture has become white and thick, about 5 minutes. Remove bowl from heat, add vanilla extract and continue beating until stiff, glossy peaks begin to form, about 3 minutes.
Spray the inside of a piping bag with non-stick spray, then scoop frosting mixture into piping bag and twist the top closed. Cut off the end of the piping bag, about an inch from the end, then pipe a dollop of frosting onto each cooled cupcake.
Preheat oven to broil, then place frosted cupcakes onto a cookie sheet and place on top rack in oven. Watch cupcakes closely as the frosting begins to turn golden. Remove cupcakes from oven. Cool slightly then enjoy immediately or keep at room temperature until ready to serve.