1. In bowl of blender, add evaporated milk, cream of coconut, sweetened condensed milk, coconut milk, rum (if using), vanilla extract and ground cinnamon. Blend on high until mixture is well combined, 1-2 minutes.
2. Pour coconut mixture into glass bottles; cover. Transfer to refrigerator. Chill until cold.
3. To serve, pour coquito into small serving glasses. Garnish with ground cinnamon and cinnamon sticks, if desired.