1. Combine coconut and sugar. Stir in egg white, water and 1/2 teaspoon vanilla. Form into a 16-inch roll. Wrap in plastic wrap; chill.
2. Cream butter; add brown sugar, creaming well. Blend in unbeaten yolk, 1/2 teaspoon vanilla and salt. Beat well. Add flour; mix well. Form into a 10-inch roll Wrap in plastic wrap; chill.
3. Cut cookie dough into 1/4-inch slices. Place on ungreased cookie sheets. Cut coconut roll with a sharp knife into slices, about 1/4-inch thick; shape into patties the same size as the cookie rounds. Place on cookie slices.
4. Bake at 375 degrees F. for 12 to 15 minutes until golden brown. Melt the chocolate morsels. Frost cookies. To make a petal design in the frosting, use tines of a dinner fork. Cut cherries in sixths. Place one on each cookie.