BIALYSTOK, Poland—Distillers here have the American spirit—vodka from where the buffalo roam. But this cocktail has a twist: It's banned in the U.S.

Now, after nearly a decade of work on two continents to formulate and brand a legal version of the alcohol, its producers are taking a shot at the American market. The booze, called Żubrówka, is unusual because it is flavored with a rare, pungent wild grass enjoyed by European bison. Each bottle has a blade of the grass in it for the drinker to admire.

Bison...