Brink milk almost to a boil in a medium saucepan. Then pour in 1 cup of rice. Cook until rice is tender, about 20 minutes, stirring frequently so that the milk and rice does not burn on the pot. Turn off heat.
Beat eggs in a medium bowl. Pour 1 cup of the rice and milk mixture in the bowl with the eggs. Stir. Then pour back into pot. Turn heat on to medium. Pour in sugar and butter, stir. Cook for 3-5 minutes. Turn heat off. Stir in vanilla. Let sit for 5 minutes before serving.