After Wall Street's closing bell, on the hunt for burgers in New York

Mike Puma's hamburger cufflinks might be the only giveaway about his mission.

Seated among a large group of friends, Puma is at The Chester in Midtown Manhattan on a Tuesday night. These guys, however, are not just any group of Wall Street workers decked out in suits, ties and pocket squares.

Gotham Burger Social Club members.
Mike Puma

That—and the absence of talking shop during the Gotham Burger Social Club's monthly meeting—give onlookers a clue.

Puma is the director of investments for a Wall Street firm's private client group. He is the founder of the group, which reviews a different Manhattan restaurant's burger each month.

It all began in October 2013, "with the idea of just getting old friends together once a month," he told CNBC.

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The men, many of whom worked together at one time and most of whom are in the financial field, rate burgers along four categories (bun, patty, toppings and fries) using a scale of 1 to 10. Then, they calculate the average and post a review on Instagram.

Gotham Burger Social Club, which generally has a turnout of about 15, draws a sizable following on the picture sharing site, with more than 11,000 followers.

Don't trick out the burger; buns matter

Mike Puma’s hamburger cufflinks.
Katie Little | CNBC

So what makes a good burger?

The bun is important, said John Callea, a financial advisor for a Wall Street firm. "If it's too soggy or breaks apart, that can ruin everything."

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Fellow member John Faron, regional vice president at a large ETF provider, says a burger "should not be too tricked up," meaning the toppings should not take away from the patty itself.

The top three burgers for the group have been (in order): the "Black Label" burger at Minetta Tavern, Burger & Barrel's "Bash Burger," and the lamb burger from The Breslin, Puma said.

The group does not cherry pick toppings to customize their burgers. Instead, they opt for the chef's recommendation.

"We want to see what he created," Puma said. "We don't want to monkey with his craft. If he says it's his best burger, that's the one we want to try."

The Chester's hanger steak burger with pimento cheese, miso-Russian dressing, handmade bun and steak fries ranked 7.15 out of 10. That is a good score—but lower than the 8.75 out of 10 at Minetta Tavern.

Guys night out

The Chester’s burger with hanger steak, pimento cheese, miso-russian dressing, handmade bun.
Mike Puma

So far, the get-togethers remain a "guys' night" affair, with no female members. While they do get requests to join the club, space is limited given the group's already unwieldy size that some restaurants find hard to accommodate.

The reception to Gotham Burger has been so encouraging that members are even putting together a plan to pitch a show based on the club.

"We're trying to get Lipitor to sponsor us," added Puma jokingly.