Pour the coconut milk, heavy cream and water into a large saucepan, add the cinnamon stick and bring to a boil. Simmer and continually stir for 5 minutes. If you are using an electric stove top you may need to move your pan on and off the heat to keep it at a simmer but to keep it from spilling over. Turn off heat.
In a large glass bowl, add the egg yolks and confectioners sweetener. Whisk together.
Add a small amount of the hot cream mixture to the eggs to temper them. Whisk together and continue to add about half of the cream mixture to the tempered eggs a little at a time. Remove cinnamon stick from saucepan. Pour the tempered egg mixture back into the hot cream, whisking in slowly over low heat, in the saucepan, until completely smooth.
Continue to stir the mixture over low heat for about 5-8 minutes or until the temperature reaches 160 degrees F and coats the back of a spoon.
Remove from the heat and stir in rum and vanilla extract. Stir together until smooth. Alternately you can also place the mixture in a blender to completely combine. Be careful as it will be hot so the cover should be slightly open to allow steam out while blending.
Taste and adjust sweetener if needed, then chill for about 1-2 hours.
To serve, pour into glasses and sprinkle some ground nutmeg over the top.
Store in the refrigerator, covered, for up to 3 days. If it becomes too thick for your liking, simply whisk in some unsweetened nut milk of choice or water to thin it out a bit.