In a 6- to 8-quart pan, combine lamb, onion, carrot, 2 1/2 tablespoons tarragon, and 1 cup water. Cover and cook over high heat for 20 minutes.
Uncover and cook until liquid evaporates and drippings are browned, about 15 minutes; stir often as mixture browns.
Add wine and broth. Stir to scrape browned drippings free, then cover pan and simmer over low heat 15 minutes. Add potatoes, cover, and simmer until lamb and potatoes are tender when pierced, about 25 minutes.
Meanwhile, rinse asparagus and snap off and discard tough ends. Cut asparagus diagonally into 1-inch lengths.
When meat is tender, stir asparagus into stew. Simmer, uncovered, until asparagus is tender when pierced, about 5 minutes. Season to taste with salt and pepper and ladle into wide soup bowls. Sprinkle stew with remaining tarragon.