Milwaukee Brewers relief pitcher Jeremy Jeffress grew up in South Boston, Va., a small town just north of the North Carolina border with a population of about 8,000. As a child, he played basketball at a local park near his home, but he frequently struggled to concentrate. The smell of fish frying at Breedlove’s, wafting over from down the road, left him distracted—and hungry.

Jeffress always loved Breedlove’s, the namesake local seafood joint that Breedlove Ford started nearly four decades ago as a stand at the Halifax County...