Combine cooked rice, 1 1/2 cups of the milk (save the remaining ½ cup for adding later), and salt in a saucepan over medium heat.
Cook, stirring occasionally until thickened, around 10-15 minutes. You don’t want it to boil (or stick to the bottom of the saucepan, so keep it moving!)
Reduce the heat to medium-low, and stir remaining 1/2 cup milk, raisins, beaten egg, and sugar in, stirring.
Continue cooking 2 to 3 minutes – to allow the egg to cook and blend into the pudding.
Remove saucepan from heat; stir butter and vanilla extract into the pudding.
Devour hot, chilled or room temp.