Place the water and ¾ cup of xylitol or 1 cup erythritol in a small saucepan over a medium heat. You want the syrup to reach about 120°C. While the sweetener syrup is heating up, begin whipping the egg whites with an electric mixer. Once the egg whites have a frothy consistency, add the cream of tartar, and then add in the remaining sweetener once the egg whites start forming soft peaks. Keep whipping until the egg whites form stiff peaks.
Ideally, you want the syrup to reach the right temperature at the same time the egg whites are ready. Put the electric mixer on a low speed. Gently add the syrup into the eggs in a thin stream, but do not let the syrup touch the sides of the bowl or the whisk.
Now whisk on a high speed for roughly ten minutes until the mixture has cooled down. Reduce the speed of the mixer and start adding in the butter one tablespoon at a time until the icing is smooth and thick. If your frosting starts to look like it has curdled, just add more butter and keep mixing.
Mix in the vanilla.
You can now use as is or add your flavour of choice. Add cocoa/cacao powder to make a chocolate icing or maybe even add in some peanut butter.