1. In the heatproof bowl of an electric mixer set over a saucepan of simmering water, combine the egg whites and sugar. Cook, whisking constantly, until sugar has dissolved and the mixture is warm to the touch (about 160 degrees).

2. Attach bowl to the mixer fitted with the whisk attachment. Beat the egg-white mixture on high speed until it holds stiff (but not dry) peaks. Continue beating until the mixture is fluffy and cooled, about 15 minutes.

3. Switch to the paddle attachment. With the mixer on medium-low speed, add the butter several tablespoons at a time, beating well after each addition. (If the frosting appears to separate after all the butter has been added, beat on medium-high speed until smooth again, 3 to 5 minutes more.) Beat in vanilla. Beat on lowest speed to eliminate any air bubbles, about 2 minutes. Stir with a rubber spatula until frosting is smooth.