Preheat your oven to 350°F. Place an ungreased 10-inch tube pan nearby. Sift the cake flour and 3/4 cup of sugar together into a medium bowl three times to aerate the mixture, then set it aside.
In a large, completely clean and dry mixing bowl, beat the egg whites, cream of tartar, and salt with an electric mixer on medium speed until soft peaks form. The mixture should look foamy and hold a soft shape when the beaters are lifted.
Gradually add the remaining 3/4 cup of sugar, about 2 tablespoons at a time, while beating on high speed. Continue beating until stiff, glossy peaks form. The meringue should be thick and smooth. Gently beat in the vanilla and almond extract.
Sift about one-quarter of the flour mixture over the meringue. Use a large rubber spatula to fold it in gently until just combined. Repeat this process, sifting and folding in the flour in three more additions, being careful not to deflate the batter.
Gently spoon the batter into the ungreased tube pan. Smooth the top with a spatula. Bake for 35-40 minutes, or until the top is golden brown and the crust feels dry and springs back when lightly touched.
Immediately invert the pan onto its cooling legs or over the neck of a bottle. Let the cake cool completely upside down for at least 1 1/2 hours to prevent collapsing.
Once completely cool, run a long, thin knife or a metal spatula around the edges of the pan and the center tube to loosen the cake. Invert the pan onto a serving plate and gently remove the cake.