What cheese made in France has a rich mushroom flavor?

The cheese you're likely thinking of is Reblochon.

Reblochon is a soft, washed-rind cheese from the Savoie region of France. It's known for its intense, earthy flavor that can indeed be reminiscent of mushrooms. This flavor comes from the combination of its pasteurized cow's milk, the specific bacteria used in the aging process, and the use of local herbs and spices.

Reblochon is often enjoyed melted in dishes like tartiflette (potatoes, bacon, and onions), but it's also delicious on its own.