Heat a medium-sized pot over medium heat. Melt the butter. Add the cinnamon stick and 1 tablespoon sugar, and stir to dissolve. Add the diced mango, and saute until the mango caramelizes slightly. Remove the mango mixture, leaving the cinnamon stick in the pan, and set the mango aside in a bowl.
In the same pan drizzle in the vegetable oil. Add the jasmine rice, and stir to coat the grains in the oil.
Pour in the whole milk, evaporated milk, water, lime zest, and remaining 2 tablespoon sugar, and stir to combine. Simmer the rice for 20 minutes. Once the pudding is cooked, season it with the nutmeg, stir and remove the heat. Add half of the mango mixture back to the pan, and mix.
To serve, ladle the pudding into a bowl and garnish with the reserved mango mixture.