Combine the sugar, cream of tartar, and water in a small saucepan fitted with a candy thermometer. Stir the sugar to dissolve and begin to heat it over medium-low. Have a heatproof measuring cup sitting nearby.
Put the egg whites and vanilla in a stand mixer fitted with the whisk attachment. When the sugar reaches 240 degrees F, immediately pour it into the measuring cup to prevent it from getting hotter. With the mixer on medium speed, slowly pour the sugar syrup into the egg whites, aiming for the side of the bowl rather than the whisk. When all the syrup is added, turn the mixer to medium-high and whisk until the icing becomes thick and holds a firm peak. Continue to whisk until the icing is just slightly warm and very thick, about 10 minutes total. Do not continue to beat, or the icing will become too thick to spread and pipe.
Use immediately on a Lime Meringue Tart, Old-Fashioned Cake, or on any other cake. Boiled icing must be used fresh and cannot be stored.
Reprinted with permission from Miette by Meg Ray with Leslie Jonath, Chronicle Books copyright (c) 2011