In a small bowl, soak 1 cup small tapioca pearls in 2 ½ cups of milk. Stir, and place in the refrigerator for 3 hours.
After letting the tapioca soak, pour the tapioca and milk into a medium saucepan. Add to the tapioca mixture: 1 ½ cups eggnog, ¼ cup sugar, salt, cinnamon, and nutmeg. Heat on medium heat, stirring frequently to keep milk from scalding. Once the tapioca mixture has almost reached a boil (do not boil), turn down the heat slightly, just to maintain a low simmer. Simmer for 10 to 15 minutes, stirring constantly to keep the milk from scalding. The tapioca should thicken a lot during this step.
Carefully taste-test tapioca to be sure pearls are soft and no longer chewy. If you need to cook it longer, cook for an additional 5 minutes and check again, repeating until done. If your tapioca is taking longer than 20 minutes, turn the heat up a little, and continue stirring.
When the tapioca is done, remove it from heat. Stir in 1 teaspoon of vanilla, and let cool for 15 minutes.
While the tapioca is cooling, beat egg whites with 1 tablespoon sugar until stiff peaks form. Fold into the tapioca, and chill in the refrigerator until ready to serve.
Serve warm or chilled for 4 hours in the refrigerator. To serve, divide tapioca into individual bowls, and top with whipped cream and a sprinkle of ground nutmeg.