Infuse the milk: Add the whole milk, cinnamon stick, and lemon peel to a large heavy pot. Warm over medium heat just until steaming and small bubbles form at the edges (do not boil). Turn off the heat, cover, and let steep 10 minutes while you start the rice.
Par-cook the rice: In a separate saucepan, bring the water to a boil. Stir in the rice and reduce to a lively simmer. Cook uncovered for 8 minutes, stirring once or twice, then drain well.
Combine and simmer: Remove the cinnamon stick and lemon peel from the milk. Return the pot to medium-low heat and add the drained rice. Maintain a gentle simmer, stirring often and scraping the bottom and corners of the pot, for 10 minutes.
Sweeten and cook to creamy: Stir in the sugar and continue cooking over low heat, stirring frequently, until the pudding looks glossy and the grains are tender with a little bite, 8–12 minutes more. It should be pourable yet thick enough to coat a spoon; it will set further as it cools.
Finish off the heat: Remove from heat and stir in the vanilla. Cover the pot and let the pudding rest 5 minutes to thicken evenly.
Serve: Spoon into bowls and dust lightly with ground cinnamon. Enjoy warm, or cool to room temperature and chill until cold for a firmer set (press plastic wrap directly on the surface to prevent a skin).