In a large pitcher with a lid, add rum and cinnamon sticks. Let soak for at least 24 hours, or up to a week.

In a blender, puree sweetened condensed milk, cream of coconut, coconut milk, evaporated milk, nutmeg, cinnamon, and vanilla extract until smooth.

Pour mixture into the bottle of rum and shake well to combine.

Chill for at least 4 hours to allow the flavors to meld and the Coquito to thicken.

Store in an airtight container in the refrigerator for up to 8 weeks. Shake vigorously before serving. Serve chilled with a little cinnamon or nutmeg on top.