SWITZERLAND HAS a number of claims to fame. Watches. Cheese. Roger Federer. The Matterhorn. Every tourist wants to see the last two, in particular, but usually settles for the mountain. And the Matterhorn delivers—as long as you don't mind seeing it with the crowds.

But it's not the 14,692-foot-high behemoth in the rugged canton of Valais that most deserves attention these days. Nor is it the dozens of other massive peaks spread across Valais, or posh ski resorts like Zermatt and Verbier. It's what fills the region's...