Preheat the oven to 350°F and position your oven rack in the center.
Sift the flour, salt, and half of the sugar together. Repeat this 4-5 times to fully combine. Set aside.
Add the egg whites to a clean metal bowl of a stand mixer fitted with a whisk attachment. Beat the egg whites at medium-low speed until slightly frothy, about 3 minutes. Add the cream of tartar and vanilla, then increase speed to medium and continue whipping until the egg whites begin to fluff up, another 1 minute.
Slowly add the remaining sugar, one tablespoon at a time. Wait about 10 seconds between each addition to allow the sugar to incorporate into the egg whites. Once all the sugar is added, increase the speed to medium-high and whip until you have stiff, glossy peaks.
Sift about 1/3 cup of the flour mixture on top and use a rubber spatula to gently fold it into the whipped egg whites. Repeat until all the flour is incorporated.
Pour the cake batter into an ungreased tube pan and smooth the top with the spatula. To remove any large air bubbles, firmly tap the pan on a counter 2-3 times.
Bake for 35-40 minutes until the top is golden brown and springs back when pressed.
Carefully remove from the oven and immediately flip the cake pan over on a bottle and allow to cool for at least 2 hours. Once cooled, run a knife around the sides and inside of the pan and lift the cake out from the mold. Then, run the knife around the bottom to release it from the base. Transfer the cake to a cake stand, bottom side up, for serving. To slice, be sure to use a serrated knife for easy cutting.