In a blender, blend together the cream of coconut, evaporated milk, rum, egg yolk (if using), ground cinnamon, vanilla extract, and grated nutmeg until smooth and well combined, which usually takes 1 to 2 minutes. Taste it and add more rum if you desire a stronger kick.
Pour the coquito into a large container with a lid, cover it, and let it chill in the refrigerator for at least 2 hours or up to 4 days. Before serving, give it a good stir to ensure the flavors are well mixed.
When ready to serve, pour the chilled coquito into small cups, garnish with a sprinkle of nutmeg, and add a cinnamon stick for an extra festive touch. Now, raise your glass and savor the holiday spirit with every sip of this delightful coquito!