Mix 1 1/2 C milk, all the rice and the salt in a medium pot.
Cook over medium heat until it becomes thick and creamy.
Beat together the egg, the remaining 1/2 C milk and the sugar.
Stir the egg mixture into the rice and milk mixture in the pot.
Cook stirring constantly until the egg is set and the mixture is thick (5-10 min)
Remove pudding from the heat and stir in the butter and vanilla.