Whip the egg whites and cream of tartar in a stand mixer on high speed.
Once the egg whites start to froth, take the ⅓ cup (66g) sugar and start gradually adding it to the egg whites, 1 tablespoon at a time, whipping for at least 30 seconds in between each addition.
Meanwhile, add the 1 cup (200g) of sugar to a saucepan with the water and put it over medium heat. If you have a candy thermometer, use it to track the temperature. Do not stir the sugar, just let it dissolve.
Once the sugar syrup reaches 240°F (115°C), it's ready (see notes if you don't have a candy thermometer). By this point, the meringue in your mixer should be at a soft peak stage.
Remove the sugar syrup from the heat and immediately pour it into the mixer in a slow, steady stream. Try to avoid hitting the side of the bowl or the whisk as you pour. Also, it's best to turn the mixer down to low speed as you add the syrup - this will keep it moving without splashing the syrup around the bowl.
Once the sugar syrup is all into the meringue, the bowl should feel hot to the touch. Continue whipping on high until the bowl feels cool.
When the bowl feels completely cool, you can begin adding the butter, one small chunk at a time. The butter must be at room temperature or it won't incorporate properly. Keep whisking the meringue on high speed while adding the butter. Let it mix in completely before adding the next chunk.
About halfway through adding the butter, your frosting is going to look awful and you will think it is ruined. Don't panic! This is normal, just keep going.
Once all the butter is incorporated, add the vanilla and salt, then whip for another minute.
You should now have a wonderful, silky smooth, buttery frosting! See the troubleshooting section above if you have any problems.
Notes
If you don't have a candy thermometer, watch for when the mixture turns completely clear - this means all the sugar has dissolved. There will be bubbles all over the surface of the syrup. To double-check, take a little bit of the syrup and drop it into some cold water - if it turns into a soft, malleable ball of sugar, then the syrup is ready. If it dissolves, it's not hot enough yet. If it turns into a hard piece of candy, then it got too hot.