Bring the water to a simmer in a medium pot over high heat.
Stir in the salt and the rice.
Allow it to come back to a boil and then turn the heat to low and simmer, covered, until the rice is tender (about 15 minutes).
Fluff the rice with a fork and stir in the milk, butter, sugar, and cinnamon.
Split the vanilla bean, scrape out the seeds, and add the seeds and the empty pod to the pot.Simmer the mixture on low heat (stirring) until slightly thickened (about 5 minutes).
Remove the pot from the heat and vigorously stir in the beaten egg.
Fish out the empty vanilla bean pod before serving.