Wrap a whole beet in foil and roast in a 350 degree oven until tender (about 1 hour). Allow to cool, then peel and roughly chop. Add to food processor with eggs and process until smooth and creamy.
Preheat the oven to 350 degrees F and line a muffin pan with cupcake liners.
In the bowl of an electric mixer, use a fork to whisk together the cake flour, brown sugar, baking powder and salt. Using the paddle attachment, beat in the the butter until well incorporated. Add the beet and egg mixture and mix until just combined (do not overmix).
Fill the cupcake molds about 2/3 of the way full and put the pan in the oven. Bake for 20-25 minutes or until a toothpick comes out clean. Remove the pan from the oven and immediately transfer the cupcakes to a wire rack to prevent the cupcakes from overcooking and becoming dry. Allow the cupcakes to cool completely.
For the butter cream, beat the egg whites with the whisk attachment until they hold stiff peaks. Heat the sugar and water in a small saucepan until it reaches 245 degrees F. With the mixer on medium speed, slowly pour the sugar mixture into the egg whites being very careful not to splash the hot sugar on yourself. When you're done adding the sugar, turn the mixer back up to high and whisk until the egg mixture has cooled down.
Add the softened butter 1 tablespoon at a time until it is fully incorporated. Add the dulce de leche then whisk until the mixture is smooth and glossy. If it's too soft, put it in the fridge for a few minutes to chill, then whisk again.
Put the frosting in a piping bag with a large round tip and frost the cupcakes in a swirling pattern.