Put a pot with 2 quarts of water on to boil. Have a large bowl of ice water ready. Cut off 1 to 1-1/2 inches from the woody bottom of each asparagus spear. Peel the bottom 3 inches. When the water boils, add the asparagus and cook for 3 minutes.
Remove the spears and plunge into the ice water until cooled, about 5 minutes. Drain well.
Heat a charcoal grill, gas grill, or broiler. Toss the olive oil, bread cubes, asparagus, and salt and pepper in a mixing bowl. Grill the asparagus and croutons until charred but not blackened, about 4 minutes.
After grilling, toss the asparagus and croutons with any oil left in the bowl.
Arrange croutons and asparagus on a plate.
With a vegetable peeler, shave parmesan over the asparagus.
Drizzle with Cilantro Dressing. Garnish with chives and additional pepper.