Special equipment: Candy thermometer
Place the whole eggs and yolks in the bowl of an electric mixer fitted with a whisk attachment. Beat on medium-high speed until the eggs are pale yellow and thick, about 5 minutes. (The eggs will hold an indent made with your finger for a few seconds.)
Meanwhile, pour the sugar and 1/3 cup water in a medium saucepan set over medium heat. Attach a candy thermometer to the side of the pan and bring to the soft-ball stage, without stirring, 240 degrees F.
Begin whipping the eggs again on medium-high speed and slowly drizzle in the hot syrup, aiming the syrup between the side of the bowl and the beaters. Continue to whip until both the bottom of the bowl and the mixture are cool, 10 to 15 minutes. The mixture should be thick, but will not form a peak at this point. Beat in the salt.
While still beating on medium-high, add the butter, a few pieces at a time. As the butter is added, the buttercream may appear to thin out some, but it will thicken into firm peaks once all of the butter is added. Beat in the vanilla extract.
Use right away or refrigerate in an airtight container overnight. The buttercream may need to be beaten again to smooth out before using.
If you plan on using any other flavoring or color, add it at the very end.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.
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