Place the eggs and egg yolks in the bowl of an electric mixer fitted with the whip attachment.
Combine the sugar and water in a medium pot and stir over medium heat until it comes to a boil. When it begins to boil, stop stirring and attach a candy thermometer to the pot (if any sugar crystals have washed up on the sides, brush them away using a pastry brush dipped in cool water). Continue to cook the syrup until it reaches 240° F (final desired temperature).
As soon as the sugar syrup hits 230° F, begin whipping the eggs and yolks on medium-high speed. The goal is to have the egg mixture become pale yellow and thick by the time the sugar reaches the 240° F.
With the mixer running, add the sugar syrup in a slow, steady stream Continue to whip on high speed until the mixture is pale and very thick and the bowl is no longer noticeably warm to the touch.
With the mixer running, gradually add room-temperature butter in 1/2 tablespoon chunks to the mixer. Continue adding and mixing until all of the butter is added and the buttercream is light and smooth.
Beat in the vanilla and mix to combine. The buttercream can be used immediately or refrigerated in an airtight container.
I always carry three kinds of hot sauce in my purse. I have a soft spot for making people their favorite dessert, especially if it's pie. My first cookbook, The Fearless Baker, is out on October 24, 2017.