Cook the rice and milk: In a medium pot over medium heat add the cooked rice, 2 cups of the milk, and the salt. Cook, stirring frequently until thick and creamy. This will take about 15 to 20 minutes. 

In a small bowl beat your egg briefly with a fork. Set aside.

Add the remaining milk and temper the egg: Once your rice pudding is thick and creamy, stir in the remaining 1/2 cup milk. Let it warm up, then scoop about ½ cup of the rice pudding mixture into a medium bowl and add the beaten egg, whisking constantly. This is tempering the egg so that it doesn’t scramble when it’s added to the pot. 

Add the egg mixture back to the pot where the rice pudding is cooking and stir well.

Add sugar: Pour in the sugar and stir well to combine. Continue cooking the pudding, stirring constantly, another 2 to 3 minutes so that the egg cooks through.

Add butter and vanilla: Remove saucepan from heat. Add butter and vanilla extract into the pudding and mix until the butter has melted. 

Cool: Let the pudding rest for 5-10 minutes, giving it time to cool and also giving the rice time to absorb more liquid. 

Serve: Scoop rice pudding into bowls and garnish with ground cinnamon if desired.