Bring 1 1/2 cups water to a boil in a saucepan; stir rice into boiling water. Reduce heat to low, cover, and simmer for 20 minutes.
In a clean saucepan, combine 1 1/2 cups cooked rice, 1 1/2 cups milk, sugar and salt. Cook over medium heat until thick and creamy, 15 to 20 minutes. Stir in remaining 1/2 cup milk, beaten egg, and raisins; cook 2 minutes more, stirring constantly. Remove from heat and stir in butter and vanilla.