Prepare a charcoal grill or preheat a gas grill to medium. Toss the asparagus with enough oil to coat and sprinkle with a little salt. Put the asparagus on the grill and cook, rolling frequently to avoid burning, until brown and crisp-tender, about 5 to 7 minutes, depending on thickness; remove. Arrange mixed greens on chilled plates or a large serving platter.
To prepare the vinaigrette: Whisk together the lemon juice, zest, chives, garlic, sugar and salt and pepper. Whisk in the oil in a slow, steady stream.
When ready to assemble the salad, toss the asparagus with enough vinaigrette to coat. Arrange on top of the mixed greens. Sprinkle with the shredded Parmesan and the snipped chives. Pass any additional vinaigrette alongside.