Across the country, Keith Miller, another franchisee, is dealing with the same problem. “What employees? We don’t have them anymore,” joked Mr. Miller, who can’t find enough workers for the three Subways he owns in Northern California.

Since 2010, fast-food jobs have grown nearly twice as fast as employment over all, contributing to the economic recovery. But rapid growth has created new problems. Some say restaurants have grown faster than demand, causing a glut of competition that is another source of pressure on business owners.

Restaurant owners are also worrying about increased immigration enforcement: Nearly one in five workers are foreign-born.