Put the small pearl tapioca in a medium bowl and cover with the water; soak 30 minutes, then drain well in a fine-mesh sieve.
In a separate bowl, whisk the large eggs and granulated sugar until pale and slightly thickened, about 1 minute.
In a heavy-bottomed saucepan, combine the whole milk, kosher salt, and the soaked tapioca pearls. Cook over medium-low heat, stirring frequently and scraping the bottom, until the pearls are mostly translucent and the milk is steaming with gentle bubbles at the edge, 12–18 minutes.
Slowly whisk about 1 cup of the hot milk-tapioca into the egg mixture to temper. Pour the tempered mixture back into the saucepan.
Cook over medium-low heat, stirring constantly, until the pudding thickens and just begins to bubble, 2–5 minutes; it should coat the back of a spoon and the pearls should be fully translucent (170–175°F). Do not boil hard.
Remove from heat and stir in the vanilla extract. Transfer to a bowl; press plastic wrap directly on the surface if chilling. Let stand 10 minutes before serving warm, or cool to room temperature then chill 2–3 hours until cold and set.
Spoon into dishes and finish with a light dusting of ground nutmeg if desired. Serve warm or cold.