When Marc and Kara Lyons take their family out for a nice dinner, they typically hand their two sons cellphones to play games and watch YouTube videos. Sometimes, the parents grab the devices back to check their own email and text messages.

But on a recent night at Hearth, an Italian-inspired restaurant in the East Village, the boys, 5 and 7, colored on blank white paper with crayons. Their parents discussed the highlights of their trip to New York from their home in Fort Lauderdale, Fla., to celebrate Mr. Lyons’s 40th birthday.

The reason for the tech-free dinner? The cellphones were stashed in a small decorative box on their table, an initiative that Marco Canora, Hearth’s chef and owner, began in November to help customers disconnect from their devices for a little bit.

Tech addiction, and what staring at smartphones all the time does to cognition, well-being and productivity, have become a top concern in Silicon Valley. Google recently announced a series of “digital well-being” updates. Apple, facing pressure from a couple of large investors, has said its next version of its mobile operating system will provide users statistics about time spent on the devices.