Sift the flour, salt, and 2/3 cup of sugar twice. Set aside.
Beat egg whites and cream of tartar on medium speed for 1 minute until foamy in the large bowl of an electric mixer fitted with a whisk attachment. Increase the speed to medium-high and gradually add remaining 1 cup of sugar, a tablespoon at a time, over the course of 1 minute. Continue to beat the egg whites until soft peaks form, usually another 3-5 minutes. Mix in the vanilla and almond extract during the last few seconds of beating.
Fold in the sifted flour and sugar mixture, about 1/3 at a time, using a rubber spatula to scrape the bottom of the bowl and 10-15 strokes each time. Turn the bowl frequently while mixing until just a few streaks of flour remain. Use 10-20 extra folding strokes after the last addition to make sure everything is evenly combined and no streaks of flour remain, but take care not to deflate the egg whites too much with vigorous stirring.
Spread the angel food cake batter into an ungreased 9- or 10-inch tube pan. Run a knife through the batter to remove air bubbles.
Bake for 35 to 40 minutes for 9-inch or 30 to 35 minutes for 10-inch cake, until cake springs back when lightly pressed on top.
Cool in the pan by turning the hot cake pan upside down over a wire rack. Let the cake cool completely in the pan, then loosen from the sides and center tube by running a sharp knife around the edges. You may need to tap the pan on the counter a few times for the cake to release.
Slice with a serrated knife and serve with whipped cream and fresh fruit.