Butter a 2-quart oven-proof dish.  Set aside.

In a medium saucepan, bring 1 cup of water to a boil.  Add 1/2 cup uncooked rice to water.  Over medium heat, cook rice until almost no water remains.  Stir often.  Remove from heat. 

Lightly beat egg and 3 tbsp. sugar together.  Set aside.

Over medium heat in a double boiler, bring milk and 1 tbsp. sugar to a gentle boil.  Stir in cooked rice.  Continue to boil until thickened.  Pudding should be jiggly and soft-set.  Remove pan from heat.  Stir 2-3 spoonfuls of hot rice mixture into egg mixture to temper the egg.  Add the egg mixture and 1 tbsp. butter to rice mixture.  Stir thoroughly.  Add salt, nutmeg and vanilla.  Stir again until evenly blended.**  Pour into buttered dish.

Combine 1 tbsp. sugar, 1 tsp. cinnamon and 1 tbsp. butter to form a soft paste.  Dot over the top of the rice pudding.  Lightly swirl into the rice pudding’s surface. Broil over high heat, six inches away from the boiler, for 2-3 minutes or until the cinnamon-sugar mixture is bubbly. Watch carefully to avoid burning the sugar.  Remove from oven.  Allow to cool slightly (about 10 minutes) before serving.  Serve warm or cold.  Store in the refrigerator covered for up to a week.