In a blender, combine the evaporated milk, sweetened condensed milk, cream of coconut, and coconut milk. Add the unsweetened cocoa powder, melted chocolate, vanilla extract, ground cinnamon, and nutmeg. Blend on medium speed for about 30 seconds, or until the mixture is smooth and creamy.

Pour in your choice of white or dark rum and blend briefly for 10-15 seconds more. Adjust the strength of your drink to your preference.

Transfer the mixture into a large glass pitcher or airtight bottles. Cover securely and refrigerate for at least 2 hours or overnight.

Before serving, gently shake the pitcher to incorporate any settled ingredients. Pour into small glasses and garnish with a cinnamon stick or shaved chocolate.