Peel the white asparagus from the head down (leaving heads intact) and trim the bottoms, tying in bunches if desired. Place it in boiling salted water and cook for about 10 minutes. Carefully take up the asparagus and place it on a plate covered with paper towels or a dish towel to absorb the excess water. Chop the dill, and press the juice from the lemon.
Arrange the warm or cold asparagus on individual dishes on beds of arugula. Drizzle liberally with freshly pressed lemon juice, a grind of pepper and fresh dill.