Trim fat from beef, wrap in foil, and place in freezer for 30 minutes to facilitate slicing.
While meat is in freezer, make sauce by combining lemon juice, wine, shallots, capers, mustard, and parsley.
Gradually whisk in olive oil by pouring in as a small stream. Add salt and mushrooms and toss.
Refrigerate for at least 20 minutes. Remove meat from freezer and with a very sharp thin knife slice on a diagonal into paper thin slices.
Place beef slices on platter or individual plates. Top with sauce with additional sauce served on the side.
Sprinkle parmesan cheese over the filet slices. Serves 2 as an entree, and 6 as an appetizer.