In the bowl of a stand mixer fitted with the whisk attachment, beat egg whites, cream of tartar, and salt at medium-low speed until soft peaks form, 5 to 6 minutes. With mixer on low speed, add sugar in a slow, steady stream, beating until combined. (This should take about 3 minutes.) Increase mixer speed to medium, and beat until thick and shiny, 10 to 12 minutes, stopping to scrape sides of bowl halfway through mixing. Rub meringue between 2 fingers to make sure it is smooth and no sugar granules can be felt. Add vanilla, and beat at medium-high speed until combined, about 30 seconds.