Step 1

Using an electric mixer on medium-high speed, beat egg yolks, egg, and salt until pale and thick, about 4 minutes.

Step 2

Meanwhile, bring granulated sugar and ¼ cup water to a boil in a small saucepan fitted with a candy thermometer over medium-high heat, stirring to dissolve sugar. Continue to cook, swirling occasionally (but not stirring), until thermometer registers 248° (this is the firm-ball stage). With motor running, gradually pour syrup into egg mixture just where the eggs meet the sides of the bowl (you want to avoid the beaters to keep sugar from splattering). Beat until eggs are light and fluffy and the outside of bowl is cool to the touch, about 5 minutes.

Step 3

Beat in butter a piece at a time, mixing until incorporated before adding the next piece. If mixture looks either soupy or curdled, continue to beat until an emulsion forms and mixture is light and thick (this could take several minutes). Mix in chocolate and vanilla.

Step 4

Do ahead: Buttercream can be made 2 days ahead. Cover and chill, or freeze up to 1 month. Bring to room temperature and beat again before using. (Thaw frozen buttercream in refrigerator overnight before bringing to room temperature.)