Whisk together 5 large egg whites and 1¼ cups sugar in the heatproof bowl of an electric mixer, over a pan of simmering water until sugar has dissolved, about 5 min. Test by rubbing the mixture between your fingers--you shouldn't feel any sugar crystals.
Transfer bowl to the mixer stand with the whisk attachment and beat on high until mixture cools and stiff peaks form, 10 to 12 min.
Reduce speed to medium-high and add 1 tsp vanilla extract and 4 sticks unsalted, softened butter a little at a time. (Buttercream may curdle but will become smooth as you continue to beat it.)
Use within a few hours or refrigerate up to a week. Before using, bring to room temperature and beat on low until smooth.