In a medium-sized, heavy-bottomed saucepan, bring the milk, rice and salt to a boil over high heat. Reduce heat to low and simmer until the rice is tender, about 20-25 minutes. Stir frequently to prevent the rice from sticking to the bottom of the pan.

In a small mixing bowl, whisk together egg and brown sugar until well mixed. Add a half cup of the hot rice mixture to the egg mixture, a tablespoon at a time, vigorously whisking to incorporate.

Add egg mixture back into the saucepan of rice and milk and stir, on low heat, for 5 to 10 minutes, until thickened, or about 160°F (71°C). Be careful not to have the mixture come to a boil at this point or it will curdle. Stir in the vanilla