Combine cooked rice, 1 1/2 cups milk, and salt in a saucepan over medium heat; cook and stir until thick and creamy, 15 to 20 minutes. Stir remaining 1/2 cup milk, golden raisins, beaten egg, and white sugar into the rice mixture; stirring continually. Continue cooking until egg is set, 2 to 3 minutes. Remove saucepan from heat; stir butter and vanilla extract into the pudding.
