Add milk, cream, sugar and salt to a saucepan over medium heat. Bring to a simmer, then stir in tapioca and reduce heat to low. Cook, stirring often, until mixture has slightly thickened, and the tapioca pearls have plumped and softened, about 45 minutes.

Add the eggs to a bowl and whisk to combine.

Add a spoonful of hot tapioca mixture to the beaten eggs, stirring well as you pour it in. Repeat with another few spoonfuls of hot tapioca, stirring after each. (This helps temper the eggs and bring them to a warm temperature without "scrambling" them.)

Add tempered egg mixture to the saucepan with tapioca and stir well to combine. Cook for 2-5 more minutes, just until thickened (keep in mind it will continue to thicken as it cools). Remove from heat and stir in vanilla.

Transfer to a serving bowl and place a piece of plastic wrap directly on the surface of the pudding (to keep it from forming a rubbery skin on the surface). Allow it to cool before serving. Serve either warm or chilled.