It was early 1955 when Ray Kroc, an ambitious milkshake equipment salesman who had bought the right to expand McDonald’s throughout the country, was opening his first restaurant, in Des Plaines, Ill.

In search of a beverage supplier, Mr. Kroc called Coca-Cola to ask about selling its sodas and reached an executive named Waddy Pratt, who ran Coke’s fountain division.

“On a Saturday morning sometime after that, Waddy goes to Des Plaines to check out this guy who claims his restaurant is going to take America by storm,” said Dick Starmann, a confidant of Mr. Kroc’s, recalling a story both men had told him many times. “He pulls up to a red-and-white-tiled shop on Lee Avenue with a yellow neon sign over it, where a guy with a hose was washing down the parking lot.”

Mr. Pratt was looking for Ray Kroc, “and the guy says, ‘You’re talking to him,’ ” Mr. Starmann said in an interview.