Preheat the oven to 325°. In a standing electric mixer, beat the egg whites at medium speed until foamy. Add the cream of tartar and salt and beat until stiff peaks form. Beat in 2 1/2 cups of the sugar, a few tablespoons at a time, until smooth and glossy, about 4 minutes.
Transfer the egg whites to a large, wide bowl. Using a fine sieve, gradually sift the flour over the egg whites, gently folding in the flour with a spatula. Fold in the vanilla and almond extracts. Scrape the cake batter into a 10-inch angel food cake pan. Using a table knife, slice through the cake batter several times to release any air bubbles. Tap the cake pan once or twice on a flat surface.
Bake the cake in the center of the oven for 20 minutes. Increase the oven temperature to 350° and bake the cake for about 35 minutes longer, until a toothpick inserted in the center comes out clean. Invert the pan onto the neck of a wine bottle and let the cake cool completely.
Meanwhile, toss the peaches with the remaining 1/2 cup of sugar and refrigerate, stirring occasionally, until juicy, for at least 1 hour or up to 4 hours.
To loosen the cake, run a thin-bladed knife around the side and tube of the pan. Unmold the cake and transfer to a platter. Cut the cake into wedges and serve with the peaches and whipped cream.