In a medium saucepan, bring 1 1/2 cups of water to a boil. Stir in rice, cover, and reduce heat. Simmer until rice is tender, about 20 minutes.
In another saucepan, combine cooked rice, 1 1/2 cups milk, sugar, and salt. Cook over medium heat, stirring often, until thick and creamy, about 15 minutes.
Stir in the remaining 1/2 cup milk and beaten egg. Cook for 2 more minutes, stirring constantly.
Remove from heat and stir in raisins, butter, and vanilla extract.
Serve warm or chilled with a sprinkle of ground cinnamon on top.