Preheat oven to 350ºF.
Cut a piece of parchment paper to fit a large baking sheet. Draw two 8-inch diameter circles with pencil on the parchment. (If two circles don't fit, make each meringue base layer on a separate baking sheet.) Turn the paper over on the baking sheet so that the pencil drawn circle side in down.
Beat the egg whites until stiff and shiny peaks form. Beat in superfine sugar a few spoonfuls at a time until stiff, shiny peaks form off the beaters and in the bowl.
Sprinkle sifted cocoa, balsamic vinegar, and chopped chocolate over the beaten egg whites in the bowl, then gently fold until everything is incorporated.
Divide and spoon the chocolate meringue between the two circles on the parchment paper, spreading it to just about ½-inch inside the pencil mark; the meringue will spread during baking We learned this the first time when our two meringue layers spread and stuck to each other. Try to make the edges ever so slightly higher (like a rim) to hold in the fillings later.
Place the baking sheets in the pre-heated oven, then turn the temperature down to 300º. Bake the meringues until they are dry to the touch, about 1 hour. Turn off the heat, prop open the oven door, and allow the meringues to cool down completely inside the oven.
The cooled meringues can be stored in an air tight container in a cool, dry place for a couple of days.