One 12-ounce can evaporated milk
One 14-ounce can sweetened condensed milk
One 14-ounce can coconut milk
One 15-ounce can sweetened cream of coconut (such as Coco Lopez or Goya)
3 cups white rum (gold rum can be substituted; see Cook's Note for low- and no-alcohol versions)
2 teaspoons pure vanilla extract
1 teaspoon ground cinnamon, plus more for garnish
2 cinnamon sticks, plus 1 more stick per serving (or 1 more stick per each additional gift bottle)
1 vanilla bean, split in half lengthwise (plus half a vanilla bean per each additional gift bottle), optional
Combine the three milks with the cream of coconut in a large blender and blend until well combined. Add the rum, vanilla extract and ground cinnamon and blend together. (If your blender is small, do this in batches and pour into a large bowl as you go.)
Pour the coquito into bottles and drop 1 cinnamon stick and vanilla bean half into each. Cover and refrigerate until very cold, at least 2 hours.
Serve straight in small glasses or over ice in larger ones. Garnish with a sprinkle of cinnamon and a cinnamon stick. Leftovers will keep tightly sealed in the refrigerator for about 1 week (shake the bottle vigorously each time before serving; see Cook's Note).
Feel free to play around with the flavors of your coquito to personalize it! A dash of almond or anise extract would be lovely. Some orange or lime zest would be a fabulous addition. You can even add some cocoa powder or thinned chocolate ganache for a chocolate version! For a low-alcohol version, replace the desired amount of rum with equal parts ice-cold coconut milk or whole milk. For a non-alcoholic or virgin coquito, replace the rum with 1 cup ice-cold coconut milk or whole milk and 1 cup ice-cold filtered water. The coconut cream solidifies in the cold, sometimes making this drink very thick after a night in the fridge. Let the coquito sit out for about 15 minutes before serving, then shake vigorously. If you're in a rush, you can run the bottle under warm tap water for a minute or so.