Their warm wide smiles would have been enough to invite us in. But as the Soru brothers open the screened door of their farm and dairy, they also offer a bottle of wine, a loaf of cheese, and a bag of flatbread. The flatbread is important to the snack—and to the brothers.

On a scalding afternoon in late June, Matteo Marchionni steps out of his house and into work boots encrusted with the dusty brown dirt of the surrounding orchard. Soon his wife, Michela Baldassarri, appears, and then his mother and father, also in their boots.

Pedra Gradl emerges from her stone farmhouse, a structure so draped with vines that they nearly engulf every visible window. She walks to a rustic picnic table where her husband, Siegfried, sits looking over the valley beneath the Apennine Mountains.

For centuries, art aficionadas have wondered about history’s most mysterious smile as well as the woman wearing it.: Who is Mona Lisa, and where was she when Leonardo da Vinci recorded the smile that has kept the world guessing?

An upbeat pop song blasts from the speakers, echoing through the empty space. At Consorzio Centro Commerciale, the new mall located directly beneath the walls of Urbino, it is 11:15 a.m. on a Saturday—prime time for shopping. Yet the mall is nearly empty.

Carlo Cleri is a man out of place. A resident of Acqualagna, a city in central Italy coveted by gourmands the world-over for its pricey truffles, Cleri’s passion is cooking weeds. While others scramble to find the $300-per-ounce fungi, Cleri combs the same countryside hunting and collecting plants most people simply step on.