Put coconut on the cake immediately after it is iced, or the frosting will become too firm, and the coconut will not adhere properly.
Heat oven to 350 degrees. Sift together 1/4 cup sugar and the cake flour three times.
In the bowl of an electric mixer fitted with the whisk attachment, whip egg whites, cream of tartar, salt, and vanilla until foamy. Add the remaining sugar; continue to whip until stiff but not dry. Sift flour mixture over egg-white mixture; gently fold until incorporated.
Pour batter into an ungreased 7-inch angel- food-cake pan. Smooth the top. Bake the cake until it is golden and springy to the touch, about 45 minutes. Invert pan onto cooling rack, and cool cake completely before removing, about 1 hour.
Place cooled cake onto serving plate, and ice with fluffy frosting. Carefully garnish with toasted coconut.