Combine cooked rice and milk in a saucepan and bring to a gentle simmer.

Cook over medium-low heat, stirring often, until slightly thickened.

Whisk eggs, sugar, cinnamon, and salt in a bowl. Slowly temper with hot milk mixture.

Pour egg mixture back into saucepan and cook gently, stirring constantly, until creamy.

Remove from heat and stir in vanilla. Serve warm or chilled.