Bring a saucepan of salted water to a boil. Place the asparagus in the water and blanch until just fork tender, yet still firm, 4 to 6 minutes. Remove the asparagus from the water and dry on paper towels.

In a skillet, melt the butter over medium-high heat. Add the mushrooms, and saute until soft and they start to give off their liquid, about 4 minutes. Add the garlic and cook for 1 minute. Remove from the heat and stir in the tarragon.

Wrap 3 asparagus stalks with a prosciutto slice to make a bundle. Repeat with remaining asparagus and prosciutto. Place on a serving platter and spoon the warm mushrooms over the top. Serve.