Place cooked rice in a medium saucepan. Add 1 1/2 cups milk and salt; cook and stir until thick and creamy, 15 to 20 minutes.
Add remaining 1/2 cup milk, golden raisins, beaten egg, and white sugar; continue cooking, stirring continuously, until well blended, 2 to 3 minutes more.
Remove saucepan from heat; stir in butter and vanilla extract.