Thanks to the coconut flavor, this cake will whisk you away to a warmer, tropical climate.
Preheat oven to 325 degrees, with rack in lower third. Sift together flour and 1/2 cup sugar into a bowl. Sift again.
Put egg whites in the bowl of an electric mixer; beat on medium speed until frothy. Add cream of tartar, lemon juice, salt, and extract; beat until soft peaks form. With mixer running, gradually add remaining cup sugar. Raise speed to medium-high; beat until stiff peaks form.
Gently transfer egg-white mixture to a large, shallow bowl; sprinkle one-third of the flour mixture over top. Using a large whisk, gently fold flour mixture into batter. Sprinkle remaining flour mixture over whites in two batches; fold until combined (do not overmix). Transfer batter to a 10-inch tube pan with a removable bottom; run a knife through batter to eliminate air bubbles. Smooth top.
Bake until a cake tester comes out clean and top is springy to the touch, 45 to 50 minutes. If pan has legs, invert and stand it on them. (Alternatively, invert pan over a narrow-necked heavy bottle.) Let cake cool completely, 1 1/2 to 2 hours.
Run a knife around edges of cake; lift out of pan. Using a serrated knife, cut cake in half horizontally; place bottom on a serving platter. Spread 2 cups coconut cream on bottom layer; set other half on top, and coat with remaining coconut cream. Top cake with coconut. Refrigerate at least 1 hour and up to 3 hours.
The cake can be baked up to 2 days ahead and stored in an airtight container at room temperature. Wait until the day you serve the cake to fill and frost it with coconut cream.