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Special equipment: candy thermometer
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Place the whole eggs and yolks in the bowl of an electric mixer fitted with a whisk attachment.
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Beat on medium-high speed until the eggs are pale yellow and thick, about 5 minutes.
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(The eggs will hold an indent made with your finger for a few seconds.)
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Meanwhile, pour the sugar and 1/3 cup water in a medium saucepan set over medium heat.
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Attach a candy thermometer to the side of the pan and bring to the soft-ball stage, without stirring, 240 degrees F.
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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.
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Continue to whip until both the bottom of the bowl and the mixture are cool, 10 to 15 minutes.
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The mixture should be thick, but will not form a peak at this point.
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Beat in the salt.
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While still beating on medium-high, add the butter, a few pieces at a time.
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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.
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Beat in the vanilla extract.
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Use right away or refrigerate in an airtight container overnight.
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The buttercream may need to be beaten again to smooth out before using.