First, put the eggs and egg yolk in the bowl of a stand mixer.  If you don't have one (I didn't until very recently), you can do this with a hand mixer instead.  It's a little easier with the stand mixer, but it's definitely doable without it.

Whip the eggs until pale yellow and frothy.  Meanwhile, cook the sugar and water in a saucepan over low heat.

When the sugar syrup has reached the softball stage (238 degrees), it's ready.

Pour the sugar syrup into the mixer while it's running.  Yes, you can do this with a hand mixer...I have!  You just can't also take a picture of it. :)

Let this continue to whip until it's cooled.  Put your hand on the bottom of the bowl to check -- it should be roughly room temperature.

While this is whipping, start melting the chocolate.  I just do it directly on the stove, on very low heat.

Start adding the butter, a tablespoon or two at a time.

This will look soupy and lumpy -- that is normal.  Add more butter, a small amount at a time, until it thickens up and looks beautifully whipped.  At this point, add the vanilla and the chocolate.  You may need to scrape down the sides because the mixer will fling the chocolate everywhere.  Then again, the sides will be coated in random bits of sugar syrup, butter, and chocolate, which is delicious to taste....