In a large pitcher or two large jars, 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 for the Coquito to thicken.

Store in an airtight container in the refrigerator for up to 8 weeks. Shake before serving.