Combine the cooked rice with 1 1/2 cups of the milk, sugar, and salt in a medium saucepan over medium heat. Cook and stir for about 15 minutes, until the mixture starts to thicken slightly and become creamy.
In a separate bowl, whisk the egg with the remaining milk. Gradually whisk 1 cup of the hot rice and milk into the egg mixture to temper the egg. Add the tempered egg mixture to the pan with the hot rice and milk, stirring to combine. Stir in the raisins, cinnamon, and nutmeg.
Continue to cook for another 2 minutes, stirring constantly, until thickened. Remove from heat and stir in the butter and vanilla.
Serve warm, at room temperature, or chill and serve cold. For an extra creamy version, mix 1 cup of sweetened whipped cream into the chilled rice pudding.
If you don't have have leftover cooked rice to make this rice pudding, bring 1 1/2 cups water to a boil in a medium saucepan. Add 3/4 cup medium grain rice, cover, and reduce the heat to a simmer. Simmer for 20 minutes until the water has been absorbed and the rice is tender.