1. Heat sugar and eggs: Bring a saucepan filled with about 2 inches of water to a simmer. Meanwhile, combine sugar, egg whites, and salt in a large heat-proof mixing bowl. Set bowl over (not in) simmering water, and whisk until whites are warm to the touch and sugar is dissolved, 2 to 3 minutes. (Heating the eggs relaxes their proteins, enabling the eggs to whip up higher and more quickly; the sugar also melts, for a silky smooth consistency.) Test by rubbing between your fingers; the mixture should feel smooth.
2. Whip meringue: Attach bowl to a standing electric mixer fitted with the whisk attachment, and beat on low speed until foamy. Add cream of tartar, and beat on medium-high speed until stiff, glossy peaks form and mixture is cooled completely, about 10 minutes.
3. Mix in butter: Reduce speed to medium-low; add butter 2 tablespoons at a time, beating to incorporate fully after each addition. Don't worry if the buttercream appears curdled at this point; it will become perfectly smooth again with continued beating. After all the butter has been incorporated, beat in vanilla.
4. Finish: Switch to the paddle attachment. Beat on the lowest speed to reduce air bubbles, 3 to 5 minutes. Let stand, covered with plastic wrap, at room temperature.