My wife really, really wanted to try my rice pudding without the raisins, so here it is. After trying this, she declared that this is the best rice pudding she's ever had. The only problem with the other recipe, in her mind, was the raisins.
Bring two cups of water to a boil. Stir in rice and reduce heat to Low. Simmer for 20 minutes, covered, stirring occasionally until the water is absorbed.
Stir half-and-half, brown sugar, cinnamon, and salt into the cooked rice. Cook on medium heat, stirring frequently to prevent sticking and clumping. Continue for about 15 minutes, until thick and creamy.
Stir in egg and milk and cook for another 5 minutes.
Remove from heat and stir in butter and vanilla extract.