In a large pot, combine rice, sugar, half and half, water, butter, and salt. Turn burner to medium-low heat and allow to simmer for about 45 minutes, or until rice is tender.
Once rice is cooked, it is time to temper the eggs. In a small mixing bowl add eggs with a little bit of water. About half a cup of water should work. Whisk to combine the eggs and water. Take a scoop of your cooked rice and milk mixture and pour it into the eggs, while whisking. Repeat once more. Slowly pour your now tempered eggs, into your rice pudding, while you are stirring.
Allow to cook over medium heat until rice pudding has thickened. Add vanilla and stir. Remove from heat and allow to sit for a minute. Serve while hot or allow to cool before serving. Enjoy!