Bring water to a boil. Add rice and salt, reduce to a simmer, cover, and cook until water is absorbed.

Increase heat to medium-low. Add milk and cook, stirring constantly, until thick and creamy.

Whisk egg in a bowl. Slowly whisk in hot rice mixture to temper the egg.

Stir tempered egg and sugar into the pot and cook briefly until thickened.

Remove from heat and stir in butter and vanilla. Serve warm or chilled.