In a medium bowl, whisk whole milk, evaporated milk and condensed milk to combine and set aside.
In a medium saucepan, bring rice, cinnamon sticks and 1 ½ cups of water to boil over medium heat. Reduce the heat to a simmer and continue to cook, uncovered, until the water has almost completely evaporated, about 10 minutes.
Stir in the cream mixture, sugar and vanilla and bring back to a boil.
Reduce the heat to a simmer and add lime zest. Stir every 2-3 minutes, to keep the mixture from scorching and sticking to the pan, for 40-50 minutes, until thick but still creamy.
Remove from the heat and discard the cinnamon sticks and lime zest. Let the pudding rest for 10 minutes.
Spoon into serving bowls and serve warm.