Over medium heat, bring water and ¾ of the sugar to a boil. Cover the pot and let boil for 2-3 minutes to wash any sugar crystals off the sides of the pot.
Remove cover and cook until the sugar syrup reaches 248° F.
Fit your stand mixer with the whisk attachment. On medium speed, whip egg whites, salt and lemon juice until foamy.
Add the reserved sugar and beat to medium-stiff peaks. Try to time this so the sugar and the whites are ready at the same time.
When the sugar has reached 248° F and the whites are at medium-stiff peaks, with the mixer on medium-high, carefully pour the sugar syrup in a thin stream down the inside of the bowl. This will prevent the syrup from hitting the whisk and getting spun around the sides of the bowl instead of into your frosting. It will also cool the syrup just a bit so you don't risk curdling your whites. Whip until cool, about 5-10 minutes.
Once the meringue is cool, add the butter, one piece at a time. Whip until each piece is completely incorporated before adding another piece. The icing might start to look a little curdled, but it should come together nicely by the time you have added all the butter. Feel the mixing bowl. It should feel somewhat cool.
If it doesn't, cool the mixture down by either putting the bowl in the refrigerator for a few minutes before continuing to whip or by rubbing the outside of the bowl with a bag of frozen peas while mixing. Guess which one I do?!
Whisk in the vanilla, and you're good to go.