In a heavy nonreactive saucepan, add water, then add sugar, corn syrup and cream of tartar. The last two help keep the sugar from crystallizing. (Candy thermometer. Non-negotiable!)
Put the pot on high heat. It's going to be there for a while. Be patient and keep your eye on it: Don't go walking away and watching TV or something.
Put the yolks and eggs in a stand mixer with the whisk attachment and turn to high. (WEEE!!!) Just let it go! Eggs will triple in volume and go to the "ribbons stage." You can't overwhip!
Wait on the sugar—looking for 235, aka "soft ball." When it happens, be ready to move quickly. Turn off the mixer and add the xanthan gum, turn to medium. Remove the thermometer from the hot sugar. Lift with two hands.
Rest the lip of the saucepan on the edge of the mixer bowl. Slowly tilt and pour the sugar in a sloooow steady stream down the side of the bowl. Don't go too fast! If you do there will be chunks of scrambled eggs in your buttercream.
Once the sugar is all in, turn the mixer to high. (WEEE!!! [AGAIN!]) Beat until cool. Gauge this by putting the inside of your wrist to the outside of the bowl. It's more accurate than your hands.
Switch out the whisk for the paddle. Next we're adding the butter. It's too heavy for the whisk and you'll end up breaking your stand mixer if you stay with the whisk.
Start cutting the butter into thin pieces—you could shave it with a cheese slicer if you'd like. Add the butter piece by piece—pain in the derrière, yes, but we're making an emulsion.
See, if you dump all the butter in all at once, the butter and eggs will never combine properly, and you'll have a "broken" buttercream. You'll be able to identify this easily—it'll be a chunky, watery, hot mess.
If your buttercream does break, you can fix it! Turn to medium high, then add a little more butter, piece by piece, until fixed. Or try adding a little guar gum! This is very strong, so add a pinch and beat for a minute, then check.
Once your butter is added, turn the mixer to medium high to add some air—10, 20 seconds at most. Quelle magnifique! It should be fluffy and make you want to eat it with your fingers.
Once you have your base, there are so many ways to flavor it! Coffee powder! Vanilla beans! Dutch cocoa! Peanut butter! Caramel! Ham! Okay, maybe not ham. Party pooper.
Congratulations! You did it! You made French buttercream! You are a god among men!