Combine. In a large saucepan, combine arborio rice, sugar, and salt. Add coconut milk and almond milk, and stir to combine.
Add Cinnamon. Tuck the cinnamon stick into the mixture.
Stir, Stir, Stir. Cook the rice pudding over medium heat or medium-low heat for about 45 minutes, stirring nearly constantly. This will prevent the pudding from sticking or burning and the stirring helps release the starch from the rice, which thickens the pudding. When finished, the pudding should be thickened and the rice should be tender (but not mushy). You want the pudding to still have some liquid, so if yours is mostly absorbed, add another splash of milk at a time and stir in.
Finish. Remove the cinnamon stick from the pudding and discard. Stir in vanilla bean paste (or vanilla extract), and taste. Add additional sweetener, salt, or vanilla, as desired.
Serve & Store. Enjoy warm or cold with your favorite toppings. Store leftover rice pudding in an airtight container 2-3 days. Keep in mind the rice will continue to absorb liquid from the pudding as it sets, so you may prefer to stir in an little extra splash of milk before chilling.