Place the cooked rice, milk, granulated sugar, vanilla, cinnamon, and salt in a saucepan. Stir well before setting the pan over medium-low heat.
Bring the mixture to a gentle simmer rather than a full boil. Cook for 18 to 22 minutes, stirring often so the milk does not scorch.
Meanwhile, melt the butter in a skillet. Add the diced apples and brown sugar, then cook for 6 to 8 minutes until tender.
Once the rice pudding is creamy and slightly thickened, fold in about three-quarters of the cooked apples. Reserve the rest for topping.
Continue cooking for 2 minutes, then remove the saucepan from the heat. The pudding will become thicker as it cools.
Spoon into bowls and finish with the reserved apples. Serve warm, or refrigerate until cold and loosen with a splash of milk if needed.
