Place a medium-sized bowl of ice water on your work space.
Cut the bottom few inches (the tough portion), off of the asparagus and discard.
Rinse and peel from just below the floret to the bottom, using a vegetable peeler. (Be very gentle as white asparagus is very fragile.)
Place a steamer in a pot with the water level just below it. Over high heat, bring the water to a boil and then add the asparagus to the rack and cover. Reduce the heat to low and steam until tender, 6 to 8 minutes.
Immediately add the steamed asparagus to the bowl of ice water and let it sit for a couple of minutes. Then remove, dry, and add them to a medium-sized mixing bowl. Set aside.
In a very small bowl, whisk the lemon juice, dill, sugar, salt and oil together. Then drizzle it over the asparagus and carefully toss to be sure it's well coated.
Serve chilled or at room temperature as a side or over fresh greens as a salad.