In a large CLEAN (grease free) metal mixing bowl, whisk egg whites and sugar over a pot of simmering water. Continue simmering the water and whisking the mixture until the sugar melts and it forms a smooth syrup, about 5 -7 minutes.
Once you have a thick, smooth syrup, remove from heat and place bowl on your mixer. Using the whisk attachment, beat on low speed for 5 minutes. Add your vanilla and increase mixer speed to high and beat for five minutes more. The frosting should form stiff peaks.
Using a spoon or a reseal-able bag with the corner snipped off, place a pile of frosting on the flat side of each red cookie. Top with the flat side of a blue cookie.