1. Preheat oven to 350°.
2. To prepare cake, lightly spoon flour into a dry measuring cup; level with a knife. Combine flour and cornstarch, stirring with a whisk.
3. Place 12 egg whites in a large bowl; beat with a mixer at high speed until foamy. Add 1 1/2 teaspoons cream of tartar and salt; beat until soft peaks form. Add 1 1/2 cups sugar, 2 tablespoons at a time, beating until stiff peaks form. Beat in 1 teaspoon vanilla and ginger. Sift 1/4 cup flour mixture over egg white mixture; fold in. Repeat with remaining flour mixture, 1/4 cup at a time.
4. Spoon batter into an ungreased 10-inch tube pan, spreading evenly. Break air pockets by cutting through batter with a knife. Bake at 350° for 35 minutes or until cake springs back when lightly touched. Invert pan; cool completely. Loosen cake from sides of pan using a narrow metal spatula. Invert cake onto a plate.
5. To prepare frosting, combine 1 cup sugar, 1/4 cup water, 1/4 teaspoon cream of tartar, and 3 egg whites in top of a double boiler. Beat at low speed of a mixer 30 seconds or until blended. Place over boiling water; beat constantly on high speed until stiff peaks form and a thermometer registers 160°, about 7 minutes. Remove from heat; beat in 1 teaspoon vanilla and coconut extract.
6. Spread frosting evenly over cake. Sprinkle cake with toasted coconut.