This New Orleans cocktail traditionally uses rye whiskey. If you can't find it, try a good-quality bourbon.
Fill 2 rocks glasses with ice to chill. In a third glass or cocktail shaker, stir together sugar and water until sugar dissolves. Add rye whiskey, bitters, and ice. Stir until chilled.
Discard ice from glasses and add 1 teaspoon anise-flavored liqueur to each. Swirl to coat bottom and sides of glasses with liqueur, then pour off extra. Strain rye mixture into glasses and garnish with lemon twists.