I like this type of mystery.

From what I understand, Paulson chips on the Short Cane mold (SCV) only start showing up around 1975. In 2005, Paulson stops selling SCV chips to the home market. You need to be a licensed gambling establishment. Even before that, by the late-1990s, they restricted the use of edge spot colors/patterns available to non-gaming licensed buyers. Paulson fantasy chips from the 1990s (cars, dogs, generals, Viva Las Vegas, etc.) share similar edge spots. Your edge spots are different from these 1990s publicly available ones.

So, I would date your Broadway chips from roughly between 1975 to 1995.

Looking at newspaper databases, I don't find a Broadway club associated with gambling in New York City during that period, but a Broadway Social Club with a significant gambling operation was raided on 3/28/1982 in New Windsor, NY.

New Windsor, in Orange County, is not that far from New York City and Long Island. Map below

Just a possibility. Not nearly enough for an ID.