Cook the rice: Combine the ⅔ cup rice and 1 ½ cups water in a medium saucepan. Stir to combine and set over medium-high heat. Bring to a boil, cover with a lid, then reduce the heat to low. Cook for about 40 minutes, or until all the water is gone and the rice is tender. Fluff with a fork and let cool uncovered. You can also refrigerate the rice in an airtight container until you’re ready to make the pudding.
Make the sweetened rice milk: Add the ¼ cup rice to a blender. Blend until finely processed, almost like cornmeal. Add the hot water and stir. Cover with the lid, but leave the keyhole open so steam can escape. Let the rice soak for at least 2 hours or up to 4 hours. Now add the brown sugar and salt, and blend on high until as smooth as possible. You should end up with about 2 ½ cups sweetened rice milk.
Add 2 cups of sweetened rice milk to the medium saucepan with the cooked rice. Set the saucepan over medium-low heat and cook, stirring occasionally, for 7 to 9 minutes, until the liquid is simmering and rice milk is thick and creamy. Taste carefully and adjust the sugar and salt to taste. Add the extra milk if you want a looser pudding . This is best hot or warm.