In a small saucepan, combine ½ cup water and the cinnamon stick. Bring to a gentle simmer, then remove from heat and let cool completely.

Discard the cinnamon stick and pour the cooled cinnamon-infused water into a blender. Add evaporated milk, sweetened condensed milk, cream of coconut, coconut milk, vanilla extract, ground cinnamon, and nutmeg.

Blend on medium speed until the mixture is smooth and frothy, about 30 seconds.

With the blender running, slowly drizzle in the dark rum. Pulse briefly to combine.

Pour the coquito into glass bottles or a large pitcher. Seal and chill in the refrigerator for at least 1 hour.

When ready to serve, give it a gentle shake, pour into small glasses, and garnish with cinnamon sticks.