In a medium, heavy bottom pot, cook milk, rice and sugar over low heat, about 2 hours, stirring often.
Once rice is cooked and pudding has begun to thicken, beat 1 Tbsp milk with 1 tsp cornstarch.
Beat 2 eggs, Beat in cornstarch mixture.
Slowly add to rice pudding mixture, stirring while adding.
Let cook over low 5 minutes.
Let cool to room temp, add vanilla.
Serve warm or cold.