Place egg whites in a large bowl; let stand at room temperature for 30 minutes.
Meanwhile, place ginger and 2 tablespoons sugar in a small food processor; cover and process until finely chopped.
In a large bowl, combine the flour, 1-1/4 cups sugar, baking powder, salt and ginger mixture. In another bowl, whisk the egg yolks, eggs, oil, water and extract. Add to dry ingredients; beat until well blended.
Add cream of tartar and remaining sugar to egg whites; beat with clean beaters until stiff peaks form. Fold into batter.
Gently spoon into three ungreased 9-in. round baking pans. Cut through batter with a knife to remove air pockets. Bake on the lowest oven rack at 325° for 22-26 minutes or until cake springs back when lightly touched.
Cool for 10 minutes before removing from pans to wire racks to cool completely.
In a large bowl, beat butter until fluffy. Add the confectionersâ sugar, milk, extract and salt; beat until smooth.
Place bottom cake layer on a serving plate; spread with 1 cup frosting. Repeat layers twice. Frost sides of cake with remaining frosting. Press coconut over top and sides of cake. Garnish with orange and lime peel. Refrigerate until serving. Yield: 16 servings.