Which cocktail is prepared by rolling its ingredients together?

The cocktail that is prepared by rolling its ingredients together is the Zombie. This drink, known for its complexity and fruity flavors, typically involves various rums and fruit juices, which are combined in a mixing technique that involves rolling. Rolling helps to blend the ingredients without overly diluting them, maintaining the vibrant flavors and character that are signature to the Zombie cocktail.

In contrast, the other listed cocktails utilize different preparation methods. The Mai Tai is generally shaken to mix and aerate the ingredients, the Sloe Gin Fizz usually involves shaking and then pouring into a glass for a refreshing finish, and the Nutty Irishman is typically built in the glass or shaken, depending on the recipe. Thus, the Zombie stands out because its unique preparation process enhances the multi-layered flavor profile characteristic of this classic tiki drink.