Preheat oven to 325°. Sift together flour and 3/4 cup sugar. Sift again and set aside.
In a large bowl or standing mixer, beat egg whites until foamy. Add cream of tartar and salt. Beat until soft peaks form. Add 3/4 cup sugar and the vanilla and almond extracts. Continue beating until egg whites are firm but not dry.
Sift one-third of the flour-sugar mixture onto the egg whites and, with a rubber or silicone spatula, gently fold the mixture into the egg whites. Add remaining flour in two batches, folding gently after each addition. Turn batter into an ungreased 10-inch tube pan and bake until browned and firm to the touch, 50 to 60 minutes.
Invert cake (in pan) on a cooling rack for at least an hour. When completely cool, run a long, thin, sharp knife between cake and pan to loosen, and remove cake.
Put candy canes in a large sealable plastic bag. Crush them into small pieces with a meat pounder, rolling pin, or the bottom of a small frying pan. Sift crushed candy with a fine-mesh strainer and reserve candy dust for another use (see Notes). Set crushed candy aside.
In a large bowl, beat cream with remaining 1/4 cup granulated sugar until soft peaks form. Frost cake with whipped cream using a spatula to form swirls and peaks. Sprinkle frosted cake with crushed candy canes. (To get candy on the sides, hold your hand about 1 in. from the cake and gently toss crushed candy at the sides.) Serve immediately, using a serrated knife to cut slices.
Note: Nutritional analysis is per serving.