With a sharp knife, cut asparagus diagonally into very thin slices and transfer to a large bowl. Halve large mushrooms. Slice mushrooms very thin and add with radishes to asparagus. Toss salad gently.
In a small bowl, whisk together lemon juice, mustard and salt. Add oil in a stream, whisking, and whisk until emulsified. Drizzle dressing over asparagus salad and toss gently. Grind pepper over salad.
Spread watercress on a platter and top with asparagus salad. With a vegetable peeler, shave 1/2 to 3/4 of Parmesan into curls over salad, reserving remaining Parmesan for another use.