1 1/2 oz. rye whiskey
1/2 oz. dry vermouth
Dash or two Maraschino liqueur
Dash aromatic bitters
Four out of the five boroughs of New York have cocktails named after them, of which the Manhattan is by far the most famous. Staten Island, somehow, is the teetotaler.
When the other three borough-cocktails are mentioned, it's usually to pan them. Embury tells us that far more Manhattans than Brooklyns are made even in Brooklyn, which, while definitely true, is perhaps rubbing it in a little too much. It's a very pleasant drink from time to time.
Among those who bother to make it, there are two schools of the Brooklyn. One simply makes it a dry Manhattan (implicitly making the sweet Manhattan coextensive with the Manhattan category). This, too, is an enjoyable cocktail, but there isn't much reason to give it its own name.
I prefer the second school, which adds Maraschino liqueur to the mix, creating a decidedly different drink. Maraschino fills out the body and adds a sweet, earthy dimension to the flavor profile.
I went with Dutch's Boomtown Bitters, previously written-up, on top. Amer Picon is often specifically indicated, in this and other cocktail recipes, but there's no need to wring your hands if you haven't got it. It's a bitters. Experiment with your own citrus or aromatic bitters until you find one that you like.
I should also note that my above recipe is approximate. The Brooklyn is a great tinkerer's drink. If you find you like it with a tablespoon of Maraschino, more power to you. These proportions should at least have you playing in in the right ballpark.