In a small bowl, whisk together the olive oil, lemon juice, shallot, and salt and pepper to taste. Let stand for 30 minutes to allow the shallot flavor to mellow. Taste and adjust the balance as needed.
Preheat the oven to 350 degrees F. Toast the walnuts on a baking sheet until fragrant and lightly colored, about 15 minutes. Let cool, then break into small pieces by hand or chop coarsely.
Cut off and discard the fennel stalks, if attached. Remove the outer layer of the bulb if it is bruised or thick and fibrous. Halve the bulb and thinly slice crosswise; you can do this by hand with a sharp knife, but a vegetable slicer, such as a mandoline, makes the task easier. Cut the apple quarters crosswise into thin slices. In a large bowl, combine the fennel, apple, walnuts, and parsley. Add the dressing and toss well. Taste and adjust the seasoning.
Pass the salad and cheeses separately.
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