In the bowl of a stand mixer fitted with the whip, combine the egg yolks and whole egg and beat on medium-high speed for about 7 minutes, or until tripled in volume and very thick, light and airy.
In a heavy-bottomed saucepan, combine the sugar and water. Clip a thermometer to the side of the pan, place the pan over medium-high heat, and bring to a boil. Boil for about 12 minutes, or until the mixture registers 250 degrees. When the sugar syrup is ready, remove it from the heat. With the mixer on low speed, carefully pour the hot syrup between the whip and the side of the bowl, taking care not to hit the whip as you pour, or the hot syrup will spatter and burn you. Continue beating on medium speed for about 3 minutes, or until the outside of the bowl is warm but not hot and the mixture is slightly cool.
Add the butter and continue to whip on medium speed for a couple of minutes, or until the butter is incorporated. Raise the speed to medium-high and beat for about 10 minutes, or until thick, smooth, shiny and well-emulsified. Add the extract, a few drops at a time, beating until the desired flavor is reached. Be careful not to overbeat, or the buttercream will be grainy.
Use immediately, or cover tightly and store in the refrigerator for up to 3 days.