Prepare the Rice:
In a medium saucepan, combine the rice and 1 1/2 cups of water.
Bring to a boil over medium-high heat.
Reduce the heat to low, cover, and simmer until the rice is tender and the water is absorbed, about 15-20 minutes.
Cook the Pudding:
Add 3 1/2 cups of milk, sugar, and salt to the cooked rice.
Cook over medium heat, stirring frequently, until the mixture thickens and the rice is very soft, about 20-25 minutes.
Finish the Pudding:
In a small mixing bowl, whisk together the remaining 1/2 cup of milk, beaten eggs, and vanilla extract.
Gradually stir this mixture into the rice pudding.
Cook for an additional 2-3 minutes, stirring constantly, until the pudding thickens further and coats the back of a spoon.
If using raisins, stir them in at this point.