Add all of the ingredients in a resealable jar or container. Shake to combine. (You won't need all of this for the salad).
For the salad:
Peel each asparagus spear (this removes the tough outer skin). Shave each spear into as many perfect ribbons as you can, avoiding the tips. Save the tips and scraps for another purpose.
Add the shaved asparagus and arugula to a large serving bowl, drizzle with a little vinaigrette, and toss to combine (be careful not add too much dressing).
Top with more black pepper, the toasted breadcrumbs, and shaved Parmesan.