“I promise, you haven’t seen anything like this,” says Nikos Fraskos, a venerable Greek icon designer and restorer, as he struggles with the keys to a chapel’s door.

I’m on Chios, a Greek island that has largely escaped the typical tourist trail. It’s not because it lacks beauty, beaches or backstory; she famously boasts Homer as her progeny and guides gleefully lead you to an ancient sanctuary to the north of Chios Town and point at an eroded rock-hewn bench, the “Teacher’s Stone”, supposedly the spot where the poet held court.

No, the reason for the relative lack of package holidays is because the island’s wealth isn’t totally dependent on tourism. Chiots have always been well-off, because...