Place the Pecorino in a large bowl. While whisking, slowly drizzle in 1/2 cup of boiling water. Then, still whisking, slowly drizzle in the olive oil. Taste and season with salt and pepper.
Slice the tips off the asparagus and slice each one lengthwise in half. Use a vegetable peeler to shave each asparagus stalk into long thin ribbons (you’ll need to press hard - try placing the asparagus flat and peeling that way). Place the asparagus in a large bowl and toss with several tablespoons of the dressing. Add the arugula and toss to coat. Season to taste with more salt and pepper as needed.