Remove butter from the refrigerator and cut into tablespoon-sized pieces. Leaving the butter out while you work on the next few steps gives it just the right amount of time to soften slightly (touching it will leave a tiny indentation) while still ensuring that it is quite cold.
Place egg whites, sugar, and salt in a double boiler or the bowl of a stand mixer over a pot of boiling water.
Whisk constantly until the eggs reach 160 F according to a candy thermometer (the temperature necessary to kill salmonella in raw eggs).
Remove from heat and, in a stand mixer fitted with the whisk attachment, whisk on high speed for ten minutes or until the bowl cools to room temperature. If the bowl feels at all warm, keep whisking.
Switch to the paddle attachment and, on medium speed, beat in the butter one tablespoon at a time.
Turn the mixer up to high speed and keep beating until the frosting is light and fluffy. During this process, the gorgeous meringue will suddenly deflate and may look soupy or curdled. Just keep mixing and eventually it will come together again and become beautiful frosting (this can take up to ten minutes).
Mix in vanilla.
Notes
Use immediately, refrigerate in an airtight container for up to one week, or freeze for up to 2 months. Allow refrigerated or frozen buttercream to come to room temperature and then beat until it becomes light and fluffy again before spreading or piping on cakes. Store cakes frosted with Swiss meringue buttercream in the refrigerator, but always bring to room temperature before serving.