In top of double boiler, combine rice and 4 cups of milk. Cook over boiling water, stirring occasionally , 45 min, until rice is tender (add water to lower part of boiler if needed). Remove from hot water and cool for 10 min.
Preheat oven to 350º. Lightly butter a 2 quart casserole or baking dish, Place in a shallow baking pan.
In a large bowl, with a whisk, beat eggs with 1 cup milk. Add ¾ cup sugar, the salt and vanilla; mix well. Gradually stir in the rice mixture, and mix well. Pour into prepared casserole dish. Pour hot water to 1” depth around casserole dish. Bake, uncovered, 50-60 minutes or until knife inserted about 1” from edge comes out clean. Remove the casserole dish from hot water to a rack. Cool at room temperature for 1 hour. Then refrigerate until ready to serve (this can be overnight).
Before serving, add sugar to a snall skillet. Cook slowly over very low heat, stirring occasionally with wooden spoon, just until sugar melts and turns golden (Be careful not to cook for too long or at too high a temperature, as it become to dark and taste burnt). Remove from heat and cool slightly. Using a teaspoon, drizzle syrup over top of pudding, Pudding can be served with whipped cream.