In a large sauce pot on the stove top, heat milk, half & half and salt over medium low heat. Stirring to avoid scorching. Stir in rice and let come to a very low boil, stirring frequently. Cook rice until it becomes tender. About 10 minutes. Stir in brown sugar.
In a small bow, beat egg, cinnamon, nutmeg and vanilla. Add ½ C of rice mixture to egg and whisk immediately to temper the egg. Stir in tempered egg mixture to rest of rice. Continue to stir and cook rice until thick and creamy. Add extra half & half if more liquid is needed while cooking.
When thick and creamy, remove from heat and stir in butter until melted. Add extra salt if needed. Serve warm.