Soak rice and cinnamon sticks in water in a heavy saucepan for 30 minutes.
After soaking, bring the rice mixture to a boil on high heat, uncovered. When it starts to boil (about 5 minutes), lower the heat to medium and cook for 10 to 12 more minutes or until water is almost evaporated.
While rice is cooking, beat the egg in a bowl using a whisk, add milk, condensed milk, dulce de leche and vanilla extract. Stir well to mix.
Add the egg mixture to the rice and cook over medium-low heat, stirring carefully, until it thickens slightly or until desired consistency, about 25 to 35 minutes.
Let cool uncovered.
Serve hot or cold and sprinkle some cinnamon powder on top.
Be aware that as the rice cools, it thickens. The finished product will be thinner than traditional rice pudding