Rice Pudding

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In a medium saucepan, combine cooked rice, 2 cups milk, sugar, and salt.
Cook over medium heat until thick and creamy, about 15 minutes, stirring frequently.
In a small bowl, whisk together eggs and remaining ½ cup milk. Gradually add the egg mixture to the saucepan, stirring constantly.
Continue to cook for 2 minutes, stirring constantly.
Remove from heat and stir in butter, vanilla extract, and nutmeg.
Let cool slightly before serving. Serve warm or chilled.