When we think of cocktails, we typically think of lightly sweet concoctions that function as adults-only desserts. Rich curries and roasted mushrooms are not what come to mind. That being said, savory cocktails—drinks that feature herbs and spices more commonly found in a main course—are attracting a lot more attention in craft-cocktail dens throughout the United States.

The majority of these cocktails are descendants of the Bloody Mary or the bullshot (vodka mixed with beef consommé), but most are far more…