In a world that seems consumed by the trials of capitalism and the unyielding pressure to work, Robin Leach, the British television personality who died in 2018, noted one place that did it differently: In Italy, Mr. Leach said, they add work and life onto food and wine.

Luciano Fregonese, the mayor of Valdobbiadene — a rural town of 10,000 people in Italy’s picturesque northeast, famous for its Prosecco — has lived such truths.

“It’s not easy, because every weekend there is one or two or three or maybe more events to attend to, and every event it is common to drink wine, or Prosecco, and also to eat,” the mayor said in an interview, with the help of a translator. “It’s not easy to say ‘no.’”

Such are the burdens of a three-term mayor in this lush region of Italy, where Mr. Fregonese — concerned about the toll the job was taking on his health — has added a new assignment to his regular mayoral duties: On Thursdays, Mr. Fregonese walks.

It began as a lighthearted gag between friends, that the boisterous 47-year-old had taken his mayoral job to heart — and waistline. Mr. Fregonese, who says he has gained weight since first being elected mayor in 2014, was already becoming more concerned about his job’s toll on his health before he won a third term in June.