Combine the milk, sugar, butter, vanilla sugar, lemon zest and cinnamon stick in a saucepan over a medium-high heat.
Heat until small bubbles appear along the edge of the pan, then remove and leave to cool for 30 minutes.
Meanwhile, in another saucepan, bring the water to a boil. Add the salt and rice, lower the heat and cook slowly for 15–20 minutes until the rice kernels have swelled and are tender. Then drain the rice.
Place the saucepan containing the milk mixture over a medium heat and bring it to a simmer. Add the rice and, stirring often, leave it to simmer uncovered for about 15–20 minutes until thickened.
Remove the lemon zest and cinnamon stick. Put the hot rice pudding into bowls and serve sprinkled with cinnamon.