This weekend, I tried again, using a recipe created by Rachael Rappaport of Coconut and Lime. And, this time, I finally got the flavor and texture I'd been hoping for.
Once I located right recipe, and found the sticky rice, the dish came together easily. The recipe does require a little time, because the rice needs to be soaked for one hour before it is cooked, but I worked on other projects while the rice was soaking. The only difficult part of the process was restraining myself from drinking the sweetened coconut milk before it was combined with the rice.
If you can't find good mangoes, I think the sweetened rice would also be good with fresh peaches or nectarines.