Add one cup sugar, 1/2 tsp salt, 1/4 tsp ground nutmeg, and 1/3 cup cornstarch into a large sauce pan. Sift or stir the ingredients together, then pour in 3 cups of milk, and stir until well combined.
Place the pan over medium heat, and cook for about 15 minutes. stir continuously.
Reduce the heat to low, and scoop out about 1/2 cup of the hot milk mixture.
In a large bowl, beat three eggs. Slowly pour in the the 1/2 cup of the hot milk mixture, and continue to stir. By doing this, you are slowly bringing the eggs up to temperature, and this prevents the eggs from cooking when you add it to the sauce pan.
Return back to the stove top, and turn the heat back to medium.
Stir the hot milk mixture, and start adding in the eggs.
Stir continuously, and and add in the vanilla extract.
Stir, and cook for an additional 2 minutes.
Remove from the heat, and let it sit for 2 minutes.
Pour the hot pudding into a heat resistance bowl.
Cover the pudding with plastic wrap. Make sure that the plastic wrap is touching the pudding. This prevents the pudding from forming a film on top.
Let the pudding sit until it reaches room temperature.