Preheat oven to 400°F. Line a 15x10-inch baking pan with parchment paper. In a medium bowl toss together mushrooms, wine and 2 teaspoons tarragon; set aside.
Snap off and discard woody bases from asparagus. Place asparagus in a prepared baking pan. Drizzle with oil and sprinkle with salt and pepper. Toss to coat.
Roast, uncovered, for 5 minutes. Add mushroom mixture to the pan; toss gently to combine. Return to oven; roast about 10 minutes more or until asparagus is crisp-tender. If desired, garnish with additional fresh tarragon.