2 1/2 cups sugar
10 large egg whites
4 cups unsalted butter, cut into pieces
2 teaspoons pure vanilla extract
Place sugar and egg whites in the heat-proof bowl of an electric mixer. Set bowl over a pan of gently simmering water, and whisk until sugar has dissolved and egg whites are hot to the touch, about 3 minutes. Test by rubbing the mixture between your fingers; it should feel completely smooth.
Transfer bowl to mixer stand. Using the whisk attachment, beat on high speed until mixture has cooled completely and formed stiff and glossy peaks, about 10 minutes.
Add the butter, one piece at a time, and beat until incorporated after each addition. Don't worry if the buttercream appears curdled after all the butter has been added; it will become smooth again with continued beating. Add vanilla, and beat just until combined.
Switch to the paddle attachment, and beat on the lowest speed to eliminate any air pockets, about 5 minutes. If using buttercream within several hours, cover bowl with plastic wrap, and set aside at room temperature in a cool environment. Or transfer to an airtight container, and store in the refrigerator, up to 3 days. Before using, bring buttercream to room temperature, and beat on the lowest speed with the paddle attachment until smooth, about 5 minutes.