Heat a medium, heavy skillet (not nonstick) over high heat for 2 minutes.
Coat pan with cooking spray. Immediately add asparagus pieces to pan, shaking them into a single layer; cook, without stirring, 2 minutes or until asparagus is very lightly charred. Cook asparagus 5 more minutes or until crisp-tender and evenly charred, tossing occasionally.
Remove pan from heat. Let asparagus rest 1 minute. Add walnut oil; toss to coat asparagus pieces. Add lemon juice; toss. Turn on heat if necessary to evaporate most of liquid. Sprinkle asparagus with tarragon and salt; toss. Sprinkle with chopped walnuts and shaved Parmigiano-Reggiano. Serve immediately.