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Preheat oven to 350 degrees. Place flour and salt in a sieve set over a bowl; sift into bowl, and set aside. With a mixer, beat egg whites on medium-high until foamy, about 1 minute. Add cream of tartar; beat until soft peaks form. Continue to beat, gradually adding sugar; beat until stiff peaks form, about 2 minutes. Add vanilla; beat to combine.
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Gently transfer egg-white mixture to a large, wide bowl. In four batches, using the sieve again, sift flour mixture over egg-white mixture. While turning the bowl, use a rubber spatula to fold in the mixture by cutting down the center and coming up the sides.
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Gently spoon batter into an ungreased angel food cake pan with a removable bottom; smooth top. Cut a knife or small spatula through batter to release air bubbles. Bake until cake is golden and springs back when lightly pressed, 35 to 40 minutes. Invert pan; let cool in pan, 1 hour. Run a knife around the inside of the pan and around the tube to release cake, and unmold. Use knife to release cake from bottom of pan, and remove.
Variations
Craving chocolate? Replace 1/4 cup cake flour with 1/4 cup unsweetened cocoa powder; reduce vanilla extract to 1 teaspoon.