How to Make It
Light a grill. In a small bowl, mix the 2 tablespoons of olive oil with the soy sauce, lemon juice, vinegar and tarragon and season with salt and pepper.
Brush the shiitake with 2 tablespoons of the soy vinaigrette; season with salt and pepper. Grill over moderate heat, turning once, until just tender, about 6 minutes. Transfer the shiitake to a bowl; cut any large shiitake into quarters. Add 4 tablespoons of the soy vinaigrette and toss to coat.
Bring a large skillet of salted water to a boil. Fill a large bowl with ice water. Add the asparagus to the skillet and cook until crisp-tender, about 3 minutes. Transfer to the ice water to cool. Drain and pat dry with paper towels.
Arrange the asparagus on a platter. Drizzle with olive oil and season with salt and pepper. Spoon the shiitake over the asparagus, drizzle any remaining vinaigrette on top and serve right away.