Combine 1 cup sugar and 1/2 cup water in a 9-inch cakepan; cook over low heat 10 minutes or until sugar caramelizes, tipping pan to cover bottom evenly.
Whisk together eggs, yolks, and remaining 1/2 cup sugar until blended. Gradually add milk, whisking constantly; stir in coconut, rum, and vanilla. Pour mixture into prepared pan. Cover with foil; place on a jellyroll pan. Add hot water to jellyroll pan to a depth of 1/4 to 1/2 inch.
Bake at 325° for 1 hour and 10 minutes or until a knife inserted in center comes out clean. Remove from water, and uncover; cool in cakepan on a wire rack. Cover and chill 3 hours.
Run a knife around edge of custard to loosen; invert onto a serving plate. Garnish, if desired.
Note: 8 (4-ounce) custard cups may be substituted for cakepan. Caramelize sugar in heavy saucepan instead of cakepan.
