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...