Whisk the egg whites, sugar and salt in a heatproof bowl set over a saucepan of simmering water (do not let the bowl touch the water) until the mixture is warm and the sugar dissolves. Remove the bowl from the pan; let cool slightly.
Beat the egg white mixture in a stand mixer fitted with the whisk attachment (or with a hand mixer) on medium-high speed until stiff peaks form, 12 to 15 minutes. Beat in the butter a few pieces at a time, then continue beating until smooth.
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In a large bowl, combine sugar, shortening, water and vanilla. Beat on low speed to combine, then beat on medium speed for a full five minutes. It won't look like Icing at first, but keep the mixer going for a full five minutes, and then you're done!
Note: If you're not using this for decorating, but just for icing the cake, thin the icing by adding 3 tbs. of corn syrup, or water to the icing.