Combine water and rice over high heat in a saucepan and bring to a boil. Reduce heat to medium-low, cover, and simmer until tender, about 45 minutes.
Combine cooked rice, 1 1/2 cups milk, sugar, and salt in a clean saucepan. 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 extract. Serve warm.