1. Put egg whites and sugar in the heatproof bowl of an electric mixer set over a pan of simmering water. Whisking constantly, cook until sugar has dissolved and mixture is warm (about 160 degrees).

2. Attach bowl to a mixer fitted with the whisk attachment. Beat egg white mixture on high speed until it forms stiff (but not dry) peaks. Continue beating until fluffy and cooled, about 6 minutes.

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