His first day in Brooklyn, Joon Yoon was baptized in true New York City fashion—with bird poop. While others would consider this an ill omen, Joon saw his unexpected baptism as a harbinger of success. “Some people say it is good luck if you get pooped on,” he explained matter-of-factly over email.

His optimism was warranted. More than two decades since his 1997 arrival in New York from South Korea, Yoon—along with his brother, Jun Yoon—now manages a small green-grocery empire. The brothers own 11 stores (including two in Greenpoint), all of which are a gentlemanly variation on the original store’s name, Mr. Kiwi at 957 Broadway in Brooklyn. They have even expanded into Queens, opening Mr. Avo this year in Long Island City.

Although now bonafide American entrepreneurs, the Yoons originally lived in a provincial capital of middling size in South Korea. Rootless and with financial difficulties, they moved to the U.S. in the late 1990s, knowing no one in the New York area. When Joon first arrived at age 23, he began working in grocery stores from the Bronx to Queens at an exhausting pace—seven days a week at 14 to 18 hours a day.

In 2006, he was faced with a choice. The Woodside grocery he worked at was closing, soon leaving him without work. Joon and his family decided to take a leap and open Mr. Kiwi, the idiosyncratic name chosen spontaneously during a road trip. In the beginning, it was hard to gain traction. “They didn’t come with a lot of money or anything… When you don’t have money, there is no one who will give you money. So, they had to start with very little product in the store. Literally, maybe a one-item-per-shelf situation,” explained Jae Lim, their office manager, over the phone.

The brother-and-father team operated the store 24 hours a day, working in shifts. Junseok Yoon, their cousin, came soon after and became an integral part of the operation. Customers appreciated the cheap produce—sourced from Hunts Point Market—and generous portions from their juice bar, detailed Lim.

13 years later, one store became 11. And Mr. Kiwi was joined by Mr. Coco, Mr. Piña, Mr. Melon, Mr. Lime, Mr. Berry, Mr. Mango, Mr. Lemon, Mr. Plum, and Mr. Avo. The Yoon family has even recently opened a salad bar in Bushwick.

In the past two years, the family has decided to incorporate their eclectic fruit salad under one brand name, Kicopi (a combination of the first two letters of their first three stores—Kiwi, Coco, and Piña). They have a new logo—a kiwi bird holding a pineapple and coconut—courtesy of Brooklyn Design Factory. And with office space in Great Neck, Long Island and more than 200 employees, their business has expanded beyond a simple one-store operation.

“Their success story is pretty amazing,” commented Ms. Lim.

Despite the business’s rapid growth, the stores are still true to their roots. In Greenpoint’s two outposts—Mr. Berry (892 Manhattan Avenue) and Mr. Plum (658 Manhattan Ave.) —homages to the family’s Korean background can still be seen with the likes of red bean cakes and jars of kimchi.

But what keeps customers coming back is affordability. While the prices are “not as good as in Astoria,” they are “cheap,” explained Elizabeth, a customer shopping at Mr. Plum who declined to give her full name.

Given their success, does the Yoon family plan to expand beyond Brooklyn?

“I think because we currently have all the stores in Brooklyn, we try to stay inside Brooklyn,” affirmed Lim. “We consider ourselves to be a local mom-and-pop shop.”