Do you prefer the crispy mozzarella, tempura watercress, and black garlic mayonnaise cheeseburger or the pumpkin mustard, bacon, cranberries, and sage hot dog? For something sweet, would you rather try the black sesame milkshake, the pancake and bacon frozen custard, or stick with a cold brew float? What sounds like a scene from the Culinary Institute of Paris is actually playing out at the Shake Shack over on Varick Street in Greenwich Village, Manhattan.

These strange dishes are not on the bur...