Melt butter in a medium skillet over low heat and add mushrooms. Sauté until tender, about 5 minutes. Remove from heat and allow to cool.
Fill a pot with lightly salted water and bring to a boil. Set aside a bowl of ice water. Add ferns to boiling water and simmer for 1 minute. Drain and immediately transfer to ice water to chill. Drain again.
In a medium mixing bowl, combine mushrooms, ferns, oil, 2 tablespoons walnut vinegar and 1 tablespoon balsamic vinegar. Toss well, and season to taste with salt and pepper.
Arrange equal portions of salad in center of six plates. Drizzle a little oil and balsamic vinegar in a circle around each salad, and shave a few slices of cheese on top. Serve immediately.
The New York Times