Preheat oven to 325°. Place 1/2 cup butter in a 12- x 8- x 2-inch baking dish. Place baking dish in oven, and heat until butter melts. Remove from oven, and set aside.
Beat eggs in a large bowl; gradually add sugar, beating well. Stir in milk, rice, vanilla, nutmeg, and salt. Pour into baking dish; sprinkle coconut over top. Bake, uncovered, at 325° for 1 hour. Spoon into individual serving bowls; serve hot.