1 1/3 cups (300 grams) sugar
1/3 cup water
6 egg yolks
5 sticks (1 1/4 lbs) butter, warmed to room temperature, but not melted
Separate the eggs and put the yolks in a mixing bowl. Discard or reserve the whites for another use.
Have a pastry brush and a bowl of ice water next to the stove before you begin to heat the sugar. Sugar is very hot and can be dangerous, a lot like a pot of hot oil for deep frying, so be careful not to splash yourself; this is an operation for which you might not want any children in the room. Place the sugar and water in a heavy-bottomed pan on the stove over high heat. Do not stir the sugar or shake the pan as it heats, or the sugar may crystalize and you will have to start over. As the sugar and water come to a boil, quickly and repeatedly brush ice water all around the sides of the pan just above the sugar (don't get your brush in the hot sugar) to wash any sugar crystals that form on the sides of the pan down into the rest of the sugar -- you use very cold water so that it will not completely boil away before it can run down the side of the pan into the sugar. When the sugar reaches the soft ball stage (234F-240F), take it off the heat immediately. (Here is a page with good descriptions of the stages of sugar as it is heated.)
As soon as the sugar is off heat, start whipping the egg yolks using a wire whip on a mixer. When they have become a bit pale, start adding the warm sugar while beating the eggs with the wire whip by pouring the sugar down the side of the mixing bowl. Your goal is for the sugar to get down to the eggs without it touching the wire whip, or the whip will fling it against the sides of the bowl and it won't get incorporated into the eggs.
Once all the sugar has been added to the eggs, continue mixing it to cool it until it is no longer hot enough to melt butter. Then, while still mixing with the whip, add the butter until it is completely incorporated. The buttercream is then ready to use. You can flavor or color it however you like, with food coloring dyes, liqueurs, or extracts like vanilla, almond or coffee.
Buttercream will keep in the refrigerator for about a week, or you can freeze it for several months. Bring it to near room temperature to work with it.