Preheat oven to 350 degrees. Place oven rack in center. Sift together onto a piece of parchment paper 3/4 cup sugar, flour, cocoa powder, baking soda, and salt. Add grated chocolate; set aside.
In the bowl of an electric mixer fitted with the whisk attachment, beat egg whites and cream of tartar until soft peaks form. With machine running, add the remaining 3/4 cup sugar in a slow steady stream, beating until fully incorporated and stiff glossy peaks form. Add vanilla and almond extracts; beat to combine.
Remove from mixer. Gradually add flour mixture, gently but thoroughly, folding into egg-white mixture until fully combined. Pour into a nonstick angel food?cake pan. Tap pan on counter to remove any air bubbles. Bake until cake springs back when depressed with a finger, 30 to 35 minutes. Cool, inverted, 1 hour before removing pan. Serve with chocolate hazelnut anglaise.