Combine the rice and milk: In a medium saucepan, combine the cooked rice, milk, sugar, salt, cinnamon, and nutmeg.
Cook the mixture: Cook the mixture over medium heat, stirring constantly, until the mixture comes to a boil.
Reduce the heat: Reduce the heat to low and simmer, covered, for 18-20 minutes, or until the pudding has thickened and the rice is tender.
Add the butter and eggs: Remove the saucepan from the heat and stir in the butter and eggs until well combined.
Add the vanilla extract: Stir in the vanilla extract
Chill the pudding: Pour the pudding into individual serving cups or a large serving dish. Cover and refrigerate for at least 2 hours, or until chilled.
Serve: Serve the pudding chilled, garnished with raisins or cinnamon sticks if desired.