Pour water into a saucepan and bring to a boil over medium heat; stir in rice. Reduce heat to low, cover, and simmer until rice is tender and liquid has been absorbed, about 20 minutes.

Combine cooked rice, 1 ½ cups milk, sugar, and salt in a clean saucepan. Cook over medium heat, stirring often, until thick and creamy, about 15 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 until combined.

Serve warm.