In heavy-bottomed pot, stir together whole milk, coconut milk and sugar over medium heat. Scrape seeds from vanilla bean; add seeds and pod to milk along with cardamom and rice.
Bring milk mixture to a simmer (when small bubbles appear around the edge), stir, then reduce heat to low. Cover and cook, stirring occasionally, until rice is just tender, 20 minutes.
Uncover and cook, stirring occasionally, until rice is very tender, thickened and creamy and liquid is absorbed, 10 to 15 more minutes. Remove from heat. Discard vanilla pod, then stir in butter and salt; cover and let rest 10 minutes.
Serve warm, at room temperature or chilled and topped with plum jam.
Notes
Don’t rinse the rice – the starch on the grains helps to thicken the pudding.