Don Barrett, a Mississippi lawyer, took in hundreds of millions of dollars a decade ago after suing Big Tobacco and winning record settlements from R. J. Reynolds, Philip Morris and other cigarette makers. So did Walter Umphrey, Dewitt M. Lovelace and Stuart and Carol Nelkin. Ever since, the lawyers have been searching for big paydays in business, scoring more modest wins against car companies, drug makers, brokerage firms and insurers. Now, they have found the next target: food manufacturers. More than a dozen lawyers who took on the tobacco companies have filed 25 cases against industry players like ConAgra Foods,...