Place the water, rice, salt and coconut oil in a large saucepan.
Stir to combine and bring to a boil over medium-high heat.
Reduce the heat to a simmer, and cook for about 10 to 15 minutes, or until the rice has absorbed most of the water, leaving behind no more than a bit of thick, starchy water. Be careful not to overcook the rice.
While the rice is cooking, in a separate, medium-size saucepan, combine the coconut milk, milk, sugar, and cinnamon.
Cook over medium heat until the mixture is simmering (about 2 minutes). Once the rice in the separate saucepan is cooked pour the milk mixture into the larger saucepan.
Cook the rice and milk mixture over medium heat, stirring occasionally until the rice has absorbed much of the milk mixture and the entire mixture has thickened and begins to appear pudding-like, about 20-25 minutes more.
Add the vanilla.
Cool. Pudding will thicken as it cools.
For pears: Slice pears and toss in lemon juice and brown sugar. Roast on a baking sheet in 400 degree oven for 10-15 minutes, or until soft and caramelized.
This pudding is delicious served warm or cold. May be made ahead and reheated if desired.