MILAN — As Marianna Farina and her husband did some Christmas shopping on a windy night in Milan, she noticed lots of people walking around with small brown packages of cookies.

“I was curious,” she said. “Because I had heard about the cookie wars.”

She had found her way to a promotional pavilion set up to hype the introduction of Pan di Stelle Biscocrema, a new hazelnut cream-filled cookie by the venerable Italian breakfast brand, famous for its round cocoa cookies dotted with 11 white sugar stars.

About a month earlier, Nutella, the juggernaut of hazelnut spreads, had encroached on Pan di Stelle’s turf by introducing, after what the company said were 10 years and 120 million euros (about $133 million) in research and development, Nutella Biscuits. Ms. Farina had tried and liked them. Now she bit into the Pan di Stelle cookie. She liked it, too.

“It’s a tough one,” she said.

In the popular imagination, Italy is a country of ripe tomatoes, fresh pasta, virgin olive oil and other staples of the Mediterranean diet. In practice, increasingly corpulent Italians — and especially Italian children — are united by an insatiable hunger for snack food.