Step 1: Preheat your oven to 350°F (175°C) with the oven rack in the lower-middle position. Ensure your 10-inch tube pan (angel food cake pan) is clean, dry, and ungreased. Do not grease the pan.
Step 2: In a medium bowl, sift the cake flour and 1/2 cup of the granulated sugar together at least twice. Set aside. This step is crucial for an airy texture.
Step 3: In a large, impeccably clean mixing bowl (preferably metal or glass), combine the room-temperature egg whites, cream of tartar, and salt. Beat with an electric mixer on medium-low speed until the mixture is foamy and soft peaks begin to form.
Step 4: Gradually increase the mixer speed to medium-high. Slowly add the remaining 3/4 cup of granulated sugar, about 1 tablespoon at a time, allowing each addition to incorporate before adding the next. Continue beating until the meringue forms stiff, glossy peaks. The meringue should be smooth and shiny, not dry. Gently fold in the vanilla extract.
Step 5: Remove the bowl from the mixer. Sprinkle about one-third of the sifted flour-sugar mixture over the meringue. Using a large rubber spatula, gently fold the dry ingredients into the meringue using an upward motion, turning the bowl as you fold. Be careful not to deflate the air.
Step 6: Repeat with the remaining flour-sugar mixture in two more additions, folding just until no streaks of flour remain. Do not overmix, as this will lead to a dense cake.
Step 7: Gently spoon the batter into the ungreased tube pan, spreading it evenly with your spatula. Gently tap the pan on the counter a couple of times to release any large air bubbles.
Step 8: Bake for 40-45 minutes, or until the cake is golden brown, springs back when lightly touched, and a wooden skewer inserted into the center comes out clean.
Step 9: Immediately invert the pan onto its "feet" (if it has them) or onto the neck of a bottle (like a wine bottle). Let the cake cool completely upside down for at least 1.5 to 2 hours. This prevents the cake from collapsing and maintains its height.
Step 10: Once completely cool, use a thin knife or offset spatula to carefully loosen the cake from the outer edge of the pan and around the inner tube. Gently push the bottom up, then run the knife along the bottom to release the cake fully. Transfer to a serving plate.
Step 11: Slice with a serrated knife or use two forks to gently pull apart servings. Serve plain, with fresh berries, whipped cream, or your favorite fruit compote.