Rinse the rice in several changes of cool water and let drain. Combine the milk, sugar, cinnamon, orange slice, and salt in a saucepan and bring just to a boil over high heat. Add the rice to the milk mixture, reduce the heat to low, and simmer, partially covered, stirring frequently, until the rice is tender, 25–35 minutes.

Remove the saucepan from the heat, cover, and let the pudding stand, covered, for 10 minutes.

Remove the orange slice and cinnamon stick and stir in the vanilla extract. Cool slightly and spoon into a serving dish or 4 custard cups. Chill before serving, at least 3 hours or up to 24 hours.