NEW DELHI—Diksha Sharma visits her neighborhood Dunkin’ Donuts three times a week but has yet to buy its doughnuts or coffee. The 23-year-old publicist goes there for the burgers.

“I don’t think anyone would come just for a doughnut,” Ms. Sharma said, tucking into a Dunkin’ Donuts Original Tough Guy Chicken Burger. “My grandmother only recently found out what a doughnut is.”

It is a harsh truth America’s doughnut chain has had to embrace in Asia’s third-largest economy: Many Indians just don’t like doughnuts, and even the ones that like them are unlikely to buy them by the dozen.

The Massachusetts-based chain has had to radically rework its menu in India and rebrand itself through an advertising campaign to let consumers know it offers more than a Bavarian Cream and coffee. In India it now has almost as many burgers on the menu as McDonald’s . All its burgers in India, like McDonald’s, are beef-free.

Dunkin’s decision to put burgers on the menu for the first time anywhere underscores challenges international chains can face adapting to cultural differences.